Question

# The difference of x^{4}+2x^{2}-3x+7 and another polynomial is x^{3}+x^{2}+x-1. What is the

another polynomial?

## The correct answer is: Hence, another polynomial is x4 - x3 + x2 - 4x + 8.

### Answer:

○ Subtraction of polynomials.

○ Always take like terms together while performing subtraction.

○ In addition to polynomials only terms with the same coefficient are subtracted.

- Step by step explanation:

○ Given:

One polynomial: x^{4}+2x^{2}-3x+7

Difference: x^{3}+x^{2}+x-1.

○ Step 1:

○ Let another polynomial be A.

So,

(x^{4}+2x^{2}-3x+7) – A = (x^{3}+x^{2}+x-1)

A = (x^{4}+2x^{2}-3x+7) - (x^{3}+x^{2}+x-1)

A = x^{4 }+ 2x^{2 }- 3x + 7 - x^{3 }- x^{2 }- x + 1

A = x^{4}- x^{3 }+ 2x^{2}- x^{2 }- 3x - x + 7 + 1

A = x^{4 }- x^{3 }+ x^{2 }- 4x + 8

- Final Answer:

Hence, another polynomial is x^{4 }- x^{3 }+ x^{2 }- 4x + 8.

^{4}+2x

^{2}-3x+7

^{3}+x

^{2}+x-1.

○ Step 1:

○ Let another polynomial be A.

So,

^{4}+2x

^{2}-3x+7) – A = (x

^{3}+x

^{2}+x-1)

^{4}+2x

^{2}-3x+7) - (x

^{3}+x

^{2}+x-1)

^{4 }+ 2x

^{2 }- 3x + 7 - x

^{3 }- x

^{2 }- x + 1

^{4}- x

^{3 }+ 2x

^{2}- x

^{2 }- 3x - x + 7 + 1

^{4 }- x

^{3 }+ x

^{2 }- 4x + 8

### Related Questions to study

### Use the product of sum and difference to find 83 × 97.

So, 83 × 97 can be written (90 - 7) × (90 + 7)

(90 - 7) × (90 + 7) = 90(90 + 7) - 7(90 + 7)

= 90(90) + 90(7) - 7(90) - 7(7)

= 8100 + 630 - 630 - 49

= 8100 - 49

= 8051

Final Answer:

Hence, the simplified form of 83 × 97 is 8051.

### Use the product of sum and difference to find 83 × 97.

So, 83 × 97 can be written (90 - 7) × (90 + 7)

(90 - 7) × (90 + 7) = 90(90 + 7) - 7(90 + 7)

= 90(90) + 90(7) - 7(90) - 7(7)

= 8100 + 630 - 630 - 49

= 8100 - 49

= 8051

Final Answer:

Hence, the simplified form of 83 × 97 is 8051.

### Determine the gradient and y-intercept from the following equation: 4x + y = -10

Gradient is also called the slope of the line. The slope intercept form of the equation of the line is y = mx + c, where m is the slope of the line and c is the y-intercept. First we convert the given equation in this form. Further, we compare the equation with the standard form to get the slope and the y-intercept.

Step by step solution:

The given equation of the line is

4x + y = -10

We need to convert this equation in the slope-intercept form of the line, which is

y = mx + c, where m is the slope of the line and c is the y – intercept.

Rewriting the given equation, that is, keeping only the term containing y in the left hand side, we get

y = -4x - 10

Comparing the above equation with y = mx + c, we get

m = -4 ;c = -10

Thus, we get

Gradient = -4

y-intercept = -10

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c=0. Using this method, be careful to check that the equation is in general form before applying the formula.

### Determine the gradient and y-intercept from the following equation: 4x + y = -10

Gradient is also called the slope of the line. The slope intercept form of the equation of the line is y = mx + c, where m is the slope of the line and c is the y-intercept. First we convert the given equation in this form. Further, we compare the equation with the standard form to get the slope and the y-intercept.

Step by step solution:

The given equation of the line is

4x + y = -10

We need to convert this equation in the slope-intercept form of the line, which is

y = mx + c, where m is the slope of the line and c is the y – intercept.

Rewriting the given equation, that is, keeping only the term containing y in the left hand side, we get

y = -4x - 10

Comparing the above equation with y = mx + c, we get

m = -4 ;c = -10

Thus, we get

Gradient = -4

y-intercept = -10

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c=0. Using this method, be careful to check that the equation is in general form before applying the formula.

### Use the product of sum and difference to find 32 × 28.

So, 32 × 28 can be written (30 + 2) × (30 - 2)

(30 + 2) × (30 - 2) = 30(30 - 2) + 2(30 - 2)

= 30(30) + 30(-2) + 2(30) + 2(-2)

= 900 - 60 + 60 - 4

= 900 - 4

= 896

Final Answer:

Hence, the simplified form of 32 × 28 is 896.

### Use the product of sum and difference to find 32 × 28.

So, 32 × 28 can be written (30 + 2) × (30 - 2)

(30 + 2) × (30 - 2) = 30(30 - 2) + 2(30 - 2)

= 30(30) + 30(-2) + 2(30) + 2(-2)

= 900 - 60 + 60 - 4

= 900 - 4

= 896

Final Answer:

Hence, the simplified form of 32 × 28 is 896.

### The sum of two expressions is x^{3}-x^{2}+3x-2. If one of them is x^{2 }+ 5x - 6, what is the

other?

- Hint:

○ Always take like terms together while performing addition.

○ In subtraction of polynomials only coefficients are subtracted.

- Step by step explanation:

Sum: x

^{3}-x

^{2}+ 3x- 2

Term: x

^{2}+ 5x- 6

○ Step 1:

○ Let the other term be A.

As given sum is x

^{3}-x

^{2}+ 3x- 2

A + x

^{2}+ 5x- 6 = x

^{3}-x

^{2}+ 3x- 2

A = x

^{3}-x

^{2}+ 3x- 2 ) - ( x

^{2}+ 5x- 6 )

A = x

^{3}-x

^{2}+ 3x - 2 - x

^{2}- 5x + 6

A = x

^{3}-x

^{2 }- x

^{2}+ 3x - 5x - 2 + 6

A = x

^{3}-2x

^{2}- 2x + 4

- Final Answer:

^{3}-2x

^{2}- 2x + 4.

### The sum of two expressions is x^{3}-x^{2}+3x-2. If one of them is x^{2 }+ 5x - 6, what is the

other?

- Hint:

○ Always take like terms together while performing addition.

○ In subtraction of polynomials only coefficients are subtracted.

- Step by step explanation:

Sum: x

^{3}-x

^{2}+ 3x- 2

Term: x

^{2}+ 5x- 6

○ Step 1:

○ Let the other term be A.

As given sum is x

^{3}-x

^{2}+ 3x- 2

A + x

^{2}+ 5x- 6 = x

^{3}-x

^{2}+ 3x- 2

A = x

^{3}-x

^{2}+ 3x- 2 ) - ( x

^{2}+ 5x- 6 )

A = x

^{3}-x

^{2}+ 3x - 2 - x

^{2}- 5x + 6

A = x

^{3}-x

^{2 }- x

^{2}+ 3x - 5x - 2 + 6

A = x

^{3}-2x

^{2}- 2x + 4

- Final Answer:

^{3}-2x

^{2}- 2x + 4.

### Use the square of a binomial to find the value. 72^{2}

^{2}can be written as (70 + 2)

^{2}which can be further written as (70 + 2)(70 + 2)

(70 + 2)(70 + 2) = 70(70 + 2) + 2(70 + 2)

= 70(70) + 70(2) + 2(70) + 2(2)

= 4900 + 140 + 140 + 4

= 4900 + 280 + 4

= 5184

Final Answer:

Hence, the value of 72^{2} is 5184.

### Use the square of a binomial to find the value. 72^{2}

^{2}can be written as (70 + 2)

^{2}which can be further written as (70 + 2)(70 + 2)

(70 + 2)(70 + 2) = 70(70 + 2) + 2(70 + 2)

= 70(70) + 70(2) + 2(70) + 2(2)

= 4900 + 140 + 140 + 4

= 4900 + 280 + 4

= 5184

Final Answer:

Hence, the value of 72^{2} is 5184.

### What is the gradient of a line parallel to the line whose equation -2x + y = -7 is:

The slope/ gradient of a line is the measure of steepness of a line. It is understood that the slope of all parallel lines in the xy plane are equal. So first, we find the slope from the given equation of a line by using the slope intercept form of a line which is y = mx + c, , where m is slope and c is the y intercept. This gradient will be equal to the gradient of any line parallel to it.

Step by step solution:

The given equation of the line is

-2x + y = -7

We convert this equation in the slope intercept form, which is

y = mx + c

Where m is the slope of the line and c is the y-intercept.

We rewrite the equation -2x + y - 7, as below

y = 2x - 7

Comparing with y = mx + c, we get that m = 2

Thus, the gradient of line -2x + y = 7 is m = 2.

We know that the gradient of any two parallel lines in the xy plane is always equal.

Hence, the gradient of a line parallel to the line whose equation -2x + y = -7 is m = 2.

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c = 0. Using this method, be careful to check that the equation is in general form before applying the formula. Here, we have, a = -2, b = 1, so we get

### What is the gradient of a line parallel to the line whose equation -2x + y = -7 is:

The slope/ gradient of a line is the measure of steepness of a line. It is understood that the slope of all parallel lines in the xy plane are equal. So first, we find the slope from the given equation of a line by using the slope intercept form of a line which is y = mx + c, , where m is slope and c is the y intercept. This gradient will be equal to the gradient of any line parallel to it.

Step by step solution:

The given equation of the line is

-2x + y = -7

We convert this equation in the slope intercept form, which is

y = mx + c

Where m is the slope of the line and c is the y-intercept.

We rewrite the equation -2x + y - 7, as below

y = 2x - 7

Comparing with y = mx + c, we get that m = 2

Thus, the gradient of line -2x + y = 7 is m = 2.

We know that the gradient of any two parallel lines in the xy plane is always equal.

Hence, the gradient of a line parallel to the line whose equation -2x + y = -7 is m = 2.

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c = 0. Using this method, be careful to check that the equation is in general form before applying the formula. Here, we have, a = -2, b = 1, so we get

### Two sides of a rectangle are (3p+5q) units and ( 5p-7q ) units. What is its area?

- Hint:

The area of the rectangle is the product of sides.

- Step by step explanation:

Two sides of a rectangle

(3p+5q) units and ( 5p-7q ) units.

○ Step 1:

We know, the area of rectangle is product of its sides

i.e. area = side × side

So,

Area = (3p+5q) × (5p-7q)

= 3p (5p -7q) + 5q(5p-7q)

= 15p

^{2}- 21pq + 25pq - 35q

^{2}

= 15p

^{2}+ 4pq - 35q

^{2}sq. units

- Final Answer:

^{2}+ 4pq - 35q

^{2}sq. units.

### Two sides of a rectangle are (3p+5q) units and ( 5p-7q ) units. What is its area?

- Hint:

The area of the rectangle is the product of sides.

- Step by step explanation:

Two sides of a rectangle

(3p+5q) units and ( 5p-7q ) units.

○ Step 1:

We know, the area of rectangle is product of its sides

i.e. area = side × side

So,

Area = (3p+5q) × (5p-7q)

= 3p (5p -7q) + 5q(5p-7q)

= 15p

^{2}- 21pq + 25pq - 35q

^{2}

= 15p

^{2}+ 4pq - 35q

^{2}sq. units

- Final Answer:

^{2}+ 4pq - 35q

^{2}sq. units.

### Find the error in the given statement.

All monomials with the same degree are like terms.

- We have been given a statement in the question for which we have to find the error in the given statement.

We have given a statement all monomials with the same degree are like terms.

The above statement is not true always.

The variable should also be same.

Example:4x, 5y

Here both have degree one and both are monomials,

But since, The variables are not same they are not like terms.

### Find the error in the given statement.

All monomials with the same degree are like terms.

- We have been given a statement in the question for which we have to find the error in the given statement.

We have given a statement all monomials with the same degree are like terms.

The above statement is not true always.

The variable should also be same.

Example:4x, 5y

Here both have degree one and both are monomials,

But since, The variables are not same they are not like terms.

### Write equation of the line containing (-3, 4) and (-1, -2)

We are given two points and we need to find the equation of the line passing through them. Recall that the equation of a line passing through two points (a, b) and (c, d) is given by

Step by step solution:

Let the given points be denoted by

(a, b) = (-3, 4)

(c, d) = (-1, -2)

The equation of a line passing through two points (a, b) and (c, d) is

Using the above points, we have

Simplifying the above equation, we have

Cross multiplying, we get

2(y + 2) = -6(x + 1)

Expanding the factors, we have

2y + 4 = -6x - 6

Taking all the terms in the left hand side, we have

6x + 2y + 4 + 6 = 0

Finally, the equation of the line is

6x + 2y + 10 = 0

Dividing the equation throughout by2, we get

3x + y + 5 = 0

This is the general form of the equation.

This is also the required equation.

Note:

We can simplify the equation in any other way and we would still reach the same equation. The general form of an equation in two variables is given by ax + by + c = 0, where a, b, c are real numbers. The student is advised to remember all the different forms of a line, like, slope-intercept form, axis-intercept form, etc.

### Write equation of the line containing (-3, 4) and (-1, -2)

We are given two points and we need to find the equation of the line passing through them. Recall that the equation of a line passing through two points (a, b) and (c, d) is given by

Step by step solution:

Let the given points be denoted by

(a, b) = (-3, 4)

(c, d) = (-1, -2)

The equation of a line passing through two points (a, b) and (c, d) is

Using the above points, we have

Simplifying the above equation, we have

Cross multiplying, we get

2(y + 2) = -6(x + 1)

Expanding the factors, we have

2y + 4 = -6x - 6

Taking all the terms in the left hand side, we have

6x + 2y + 4 + 6 = 0

Finally, the equation of the line is

6x + 2y + 10 = 0

Dividing the equation throughout by2, we get

3x + y + 5 = 0

This is the general form of the equation.

This is also the required equation.

Note:

We can simplify the equation in any other way and we would still reach the same equation. The general form of an equation in two variables is given by ax + by + c = 0, where a, b, c are real numbers. The student is advised to remember all the different forms of a line, like, slope-intercept form, axis-intercept form, etc.

### Simplify and write the polynomial in its standard form.

- We have been given a function in the question.
- We will have to simplify it and further write the answer in its standard form

We know that in polynomial we add/subtract like terms

So,

Now, We know that the terms are written in descending order of their degree.

So, In the standard form

The given polynomial will be .

### Simplify and write the polynomial in its standard form.

- We have been given a function in the question.
- We will have to simplify it and further write the answer in its standard form

We know that in polynomial we add/subtract like terms

So,

Now, We know that the terms are written in descending order of their degree.

So, In the standard form

The given polynomial will be .

### What must be added to x^{3}+3x-8 to get 3x^{3}+x^{2}+6?

- Hint:

○ Always take like terms together while performing addition.

○ In addition to polynomials only terms with the same coefficient are added.

- Step by step explanation:

Sum: 3x

^{3 }+ x

^{2 }+ 6

Term : x

^{3 }+ 3x - 8

○ Step 1:

○ Let A must be added to get 3x

^{3 }+ x

^{2 }+ 6.

So,

A + x

^{3 }+ 3x - 8 = 3x

^{3 }+ x

^{2 }+ 6

A = (3x

^{3 }+ x

^{2 }+ 6 ) - (x

^{3 }+ 3x - 8)

A = 3x

^{3 }+ x

^{2 }+ 6 - x

^{3 }- 3x + 8

A = 3x

^{3}- x

^{3 }+ x

^{2 }- 3x+ 6 + 8

A = 2x

^{3 }+ x

^{2 }- 3x+ 14

- Final Answer:

^{3 }+ x

^{2 }- 3x+ 14.

### What must be added to x^{3}+3x-8 to get 3x^{3}+x^{2}+6?

- Hint:

○ Always take like terms together while performing addition.

○ In addition to polynomials only terms with the same coefficient are added.

- Step by step explanation:

Sum: 3x

^{3 }+ x

^{2 }+ 6

Term : x

^{3 }+ 3x - 8

○ Step 1:

○ Let A must be added to get 3x

^{3 }+ x

^{2 }+ 6.

So,

A + x

^{3 }+ 3x - 8 = 3x

^{3 }+ x

^{2 }+ 6

A = (3x

^{3 }+ x

^{2 }+ 6 ) - (x

^{3 }+ 3x - 8)

A = 3x

^{3 }+ x

^{2 }+ 6 - x

^{3 }- 3x + 8

A = 3x

^{3}- x

^{3 }+ x

^{2 }- 3x+ 6 + 8

A = 2x

^{3 }+ x

^{2 }- 3x+ 14

- Final Answer:

^{3 }+ x

^{2 }- 3x+ 14.

### Simplify and write the polynomial in its standard form.

- We have been given a function in the question.
- We will have to simplify it and further write the answer in its standard form

We know that in polynomial we add/subtract like terms

So,

Now, We know that the terms are written in descending order of their degree.

So, In the standard form

The given polynomial will be .

### Simplify and write the polynomial in its standard form.

- We have been given a function in the question.
- We will have to simplify it and further write the answer in its standard form

We know that in polynomial we add/subtract like terms

So,

Now, We know that the terms are written in descending order of their degree.

So, In the standard form

The given polynomial will be .

### Write the polynomial in its standard form.

- We have been given a function in the question.
- We will have to simplify it and further write the answer in its standard form.

We have given a polynomial

We know that the terms are written in descending order of their degree.

So, In Standard form

### Write the polynomial in its standard form.

- We have been given a function in the question.
- We will have to simplify it and further write the answer in its standard form.

We have given a polynomial

We know that the terms are written in descending order of their degree.

So, In Standard form

### Name the polynomial based on its degree and number of terms.

- We have been given a function in the question
- We will have to name the polynomial based on its degree and number of terms.

We have given a polynomial

Its degree is 1 and contain one variable

This is linear polynomial

### Name the polynomial based on its degree and number of terms.

- We have been given a function in the question
- We will have to name the polynomial based on its degree and number of terms.

We have given a polynomial

Its degree is 1 and contain one variable

This is linear polynomial

### Show m = 2 for the straight line 8x - 4y = 12.

We need to verify the value of m for an equation of straight line. We take the help of slope intercept form of equation of a line and convert the given equation in the form y = mx + c. Then we compare both the equations to find the value of m and check if it is equal to the given value.

Step by step solution:

The slope/ gradient of a line is denoted by m.

The given equation of the line is

8x - 4y = 12

We convert this equation in the slope intercept form, which is

y = mx + c

Where m is the slope of the line and c is the y-intercept.

We rewrite the equation 8x - 4y = 12, as below

-4y = -8x - 12

Dividing the above equation by (-4) throughout, we get

Simplifying, we have

y = 2x + 3

Comparing with y = mx + c, we get that m = 2

Thus, m = 2 for the straight line 8x - 4y = 12

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c = 0. Using this method, be careful to check that the equation is in general form before applying the formula. Here, we have, a = 8, b = -4, so we get

### Show m = 2 for the straight line 8x - 4y = 12.

We need to verify the value of m for an equation of straight line. We take the help of slope intercept form of equation of a line and convert the given equation in the form y = mx + c. Then we compare both the equations to find the value of m and check if it is equal to the given value.

Step by step solution:

The slope/ gradient of a line is denoted by m.

The given equation of the line is

8x - 4y = 12

We convert this equation in the slope intercept form, which is

y = mx + c

Where m is the slope of the line and c is the y-intercept.

We rewrite the equation 8x - 4y = 12, as below

-4y = -8x - 12

Dividing the above equation by (-4) throughout, we get

Simplifying, we have

y = 2x + 3

Comparing with y = mx + c, we get that m = 2

Thus, m = 2 for the straight line 8x - 4y = 12

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c = 0. Using this method, be careful to check that the equation is in general form before applying the formula. Here, we have, a = 8, b = -4, so we get