## Key Concepts

- List polynomial identities
- Use polynomial identities to multiply
- Use polynomial identities to factor and simplify
- Expand a power of a binomial
- Apply the binomial theorem

## Polynomial Identities

A mathematical statement that connects two polynomial expressions is an identity if one side can be transformed into the other side using mathematical operations. These polynomial identities are used to multiply and factor polynomials.

Difference of squares:

a² – b² = (a + b)(a – b)

**Square of a sum: **

(a+b)^{2} = a^{2}+2ab+b^{2}

**Difference of cube: **

a^{3}−b^{3} = (a−b)(a^{2}+ab+b^{2})

**Sum of cubes: **

a^{3}+b^{3} = (a+b)(a^{2}−ab+b^{2})

**Example:**

Find the difference of squares.

16x² – 9y²

**Solution:**

Substitute 4x for a 3y for b.

16x² – 9y² = (4x + 3y) (4x – 3y) (∵ a² – b² = (a + b)(a – b))

**Example:**

Find the square of a sum.

(36x+ 25y)²

**Solution:**

Substitute 6x for a 5y for b.

(36x + 25y)² = (6x)² + 2(6x)(5y) + (5y)²

(∵ (a + b)² = (a) ² + 2(a)(b) + (b)²)

= 36x² + 60xy + 25y²

### Use polynomial identities to multiply

The given polynomial identities are multiplied to get the required expression.

Square of a sum:

(a+b)^{2} = a^{2}+2ab+b^{2}

**Example:**

Use polynomial identities to multiply the expressions.

(12 + 15)²

**Solution:**

Find the multiplication of the expression by using the square of a sum identity.

(a + b)² = (a) ² + 2(a) (b) + (b) ²

(12 + 15) ² = (12) ² + 2(12)(15) + (15) ²

= 144 + 360 + 225

= 729

So, (12 + 15) ² = 729

### Use polynomial identities to factor and simplify

To find the factor of a polynomial identity, first, we have to find out the difference of square identity and the difference of cube identity.

Sum of cubes identity,

a^{3}+b^{3} = (a+b)(a^{2}−ab+b^{2})

**Example:**

Use polynomial identities to factor each polynomial.

12³ + 2³

**Solution:**

Use the sum cube property.

a³ + b³ = (a + b) (a² – ab + b²)

12³ + 2³ = (12 + 2) (12² – 12(2) + 2²)

= 14(144 – 24 + 4)

= 14(124)

= 1,736

So, 12³ + 2³ = 1,736

**Example:**

Find the value of *k*, if x – 3 is a factor of 5x^{3}^{ }– 2x^{2}^{ }+ x + k.

**Solution:**

Here, x – a is a factor of p(x) if p(a) = 0.

It is given that x – 3 is a factor of 5x^{3}^{ }– 2x^{2}^{ }+ x + k.

Therefore, p (3) must be equal to zero.

p(3) = 5(3)^{3 }– 2(3)^{2 }+ 3 + k = 0

Therefore,

5(27) – 2(9) + 3 + k = 0

135 – 18 + 3 + k = 0

120 + k = 0

Therefore, k = –120

### Expand the power of a binomial

The binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x+y)^{n} into a sum involving terms of the form ax^{b}y^{c} where the exponents *b* and *c* are non-negative integers with *b* + *c* = *n*, and the coefficient *a* of each term is a specific positive integer depending on *n* and *b*.

Pascal’s triangle gives us the coefficients for an expanded binomial of the form (a + b)^{n}, where *n* is the row of the triangle. The binomial theorem tells us we can use these coefficients to find the expanded binomial.

**Example:**

What is the binomial expansion of (x^{2} + 1)^{5} using the binomial theorem?

**Solution:**

The following formula derived from the binomial theorem.

**Example: **

Expand the term (1+x)^{3}

**Solution:**

We use the theorem with n = 3 and stop when we have written down the term in x^{3}

### Apply the binomial theorem

#### Binomial theorem

The binomial theorem states that for every positive integer *n*,

The coefficients

are the numbers in row *n *of Pascal’s triangle.

By using the binomial theorem, we can expand the expression and solve the problem by using Pascal’s triangle to write the coefficients.

**Example:**

Use the binomial theorem to expand the expression.

**Solution:**

Apply the binomial theorem

Here a = x and b = –1.

Therefore,

**Example:**

What is the middle term of (xyz + 3)^{80}?

**Solution:**

Since the power is even, there are an odd number of terms.

The middle term is the (n/2 + 1)^{th} term.

## Exercise

- Use polynomial identities to factor each polynomial.

m^{8 }– 9n^{10}

- Use polynomial identities to multiply the expression.

(3x^{2} – 5y^{3}) (3x^{3 }– 5y^{2})

- Use Pascal’s triangle to expand

(x-y)^{6}

- Use polynomial identities to multiply the expression.

(12-15)^{2}

- Use binomial theorem to expand the expression.

(2c-d)^{6}

- Use polynomial identities to factor each polynomial.

27x^{9} – 343y^{6}

- Use Pascal’s triangle to expand

(2x-7)^{3}

- Expand

(1+2/x)^{3}

- Expand

(1+x)^{2}

- Use Pascal’s triangle to expand

(1+p)^{4}

### Concept Map

If an equality holds true for all values of the variable, then it is called an identity.

### What have we learned

- What is polynomial identity?
- How to use polynomial identity to multiply?
- How to use polynomial identities to factor and simplify the expression?
- How to expand the power of a binomial?
- How to apply binomial theorem

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