The difference of x⁴+2x²-3x+7 and another polynomial is x³+x²+x-1. What is the
another polynomial?

83 can be written as (90 - 7) and 97 can be written as (90 + 7)
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.
 

Hint:
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.

32 can be written as (30 + 2) and 28 can be written as (30 - 2)
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.
 

Answer:

• Hint:

○      Addition of polynomials.
○      Always take like terms together while performing addition.
○      In subtraction of polynomials only coefficients are subtracted.

• Step by step explanation:

○      Given:
Sum:  x³ -x² + 3x- 2
Term: x² + 5x- 6
○      Step 1:
○      Let the other term be A.
As given sum is x³ -x² + 3x- 2
 A + x² + 5x- 6 = x³ -x² + 3x- 2
 A = x³ -x² + 3x- 2 ) - ( x² + 5x- 6 )
 A = x³ -x² + 3x - 2 - x² - 5x + 6
 A = x³ -x² - x² + 3x - 5x - 2 + 6
 A = x³ -2x² - 2x + 4

• Final Answer:

Hence, the other term is x³ -2x² - 2x + 4.

72² 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 72² is 5184.

Hint:
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 

Answer:

• Hint:

The concept used in the question is the area of a rectangle.
The area of the rectangle is the product of sides.

• Step by step explanation:

○     Given:
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)
= 15p² - 21pq + 25pq - 35q²
= 15p² + 4pq - 35q² sq. units

• Final Answer:

Hence, the area of the rectangle is 15p² + 4pq - 35q² sq. units.

Explanation:

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

Step 1 of 1:
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.

Hint:
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.

Explanation:

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

Step 1 of 1:
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 .

Answer:

• Hint:

○     Addition of polynomials.
○     Always take like terms together while performing addition.
○     In addition to polynomials only terms with the same coefficient are added.

• Step by step explanation:

○     Given:
Sum:  3x³ + x² + 6
Term : x³ + 3x - 8
○     Step 1:
○      Let A must be added to get 3x³ + x² + 6.
So,
 A + x³ + 3x - 8 = 3x³ + x² + 6
 A = (3x³ + x² + 6 ) - (x³ + 3x - 8)
 A = 3x³ + x² + 6 - x³ - 3x + 8
 A = 3x³- x³ + x² - 3x+ 6 + 8
 A = 2x³ + x² - 3x+ 14

• Final Answer:

Hence, the other term is 2x³ + x² - 3x+ 14.

Explanation:

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

Step 1 of 1:
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 .

Explanation:

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

Step 1 of 1:
We have given a polynomial 
We know that the terms are written in descending order of their degree.
So, In Standard form

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

Step 1 of 1:
We have given a polynomial 
Its degree is 1 and contain one variable
This is linear polynomial

Hint:
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