Proving Polynomial Identities

Equations that are true for every possible value of the variable are called identities. Some special polynomial equations are identities. Using polynomial identities can be useful when rewriting polynomial expressions. To prove an equation is an identity, it is enough to show that both sides can be written in the same way.

Rule

Square of a Binomial

When a binomial is squared, the resulting expression is a perfect square trinomial.

Rule

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

This identity can be shown by first rewriting the square as a product.

(a + b)^{2}

SplitIntoFactors

Split into factors

(a + b) (a + b)

MultPar

Multiply parentheses

a \cdot a + a \cdot b + b \cdot a + b \cdot b

Multiply

a^{2} + a b + a b + b^{2}

SimpTerms

Simplify terms

a^{2} + 2 a b + b^{2}

Rule

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

In this case, when one term of the binomial is subtracted from the other, the middle term of the perfect square trinomial will instead be negative.

(a - b)^{2}

SplitIntoFactors

Split into factors

(a - b) (a - b)

MultPar

Multiply parentheses

a \cdot a - a \cdot b - b \cdot a + b \cdot b

Multiply

a^{2} - a b - a b + b^{2}

SimpTerms

Simplify terms

a^{2} - 2 a b + b^{2}

Rewrite the expressions as perfect square trinomials

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Exercise

Rewrite $(x + 3)^{2}$ and $(7 - x)^{2}$ as perfect square trinomials.

Show Solution

Solution

When rewriting

(x + 3)^{2}

as a perfect square trinomial, keep in mind that it is the square of a binomial.

(x + 3)^{2}

ExpandPosPerfectSquare

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

x^{2} + 2 \cdot x \cdot 3 + 3^{2}

Multiply

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

CalcPow

Calculate power

x^{2} + 6 x + 9

The second expression has a minus sign between the terms – take this into account when rewriting the binomial.

(7 - x)^{2}

ExpandNegPerfectSquare

$(a - b)^{2} = a^{2} - 2 a b + b^{2}$

7^{2} - 2 \cdot 7 \cdot x + x^{2}

CalcPow

Calculate power

49 - 2 \cdot 7 \cdot x + x^{2}

Multiply

49 - 14 x + x^{2}

The expressions rewritten as perfect square trinomials are

x^{2} + 6 x + 9

and

49 - 14 x + x^{2},

respectively.

Rule

Product of a Conjugate Pair of Binomials

When a binomial is multiplied by its conjugate, the resulting expression is the difference of two squares.

Rule

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

This identity can be shown using the Distributive Property — multiplying each term in the first set of parentheses with each term in the second.

(a + b) (a - b)

Distr

Distribute $(a + b)$

a (a + b) - b (a + b)

Distr

Distribute $a$

a^{2} + a b - b (a + b)

Distr

Distribute $- b$

a^{2} + a b - b a - b^{2}

Simplify

CommutativePropMult

Commutative Property of Multiplication

a^{2} + a b - a b - b^{2}

SubTerm

Subtract term

a^{2} - b^{2}

Thus, the product of a binomial and its conjugate is the difference of two squares.

Rule

Sum of Two Cubes

Rule

a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})

To show the identity

a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2}),

it is enough to rewrite the right-hand side and show that it equals the sum on the left-hand side.

(a + b) (a^{2} - a b + b^{2})

MultPar

Multiply parentheses

a \cdot a^{2} - a \cdot a b + a \cdot b^{2} + b \cdot a^{2} - b \cdot a b + b \cdot b^{2}

Multiply

a^{3} - a^{2} b + a b^{2} + a^{2} b - a b^{2} + b^{3}

CommutativePropAdd

Commutative Property of Addition

a^{3} - a^{2} b + a^{2} b + a b^{2} - a b^{2} + b^{3}

SimpTerms

Simplify terms

a^{3} + b^{3}

Rule

Difference of Two Cubes

Rule

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

Showing this identity is easiest done by rewriting the right-hand side.

(a - b) (a^{2} + a b + b^{2})

MultPar

Multiply parentheses

a \cdot a^{2} + a \cdot a b + a \cdot b^{2} - b \cdot a^{2} - b \cdot a b - b \cdot b^{2}

Multiply

a^{3} + a^{2} b + a b^{2} - a^{2} b - a b^{2} - b^{3}

CommutativePropAdd

Commutative Property of Addition

a^{3} + a^{2} b - a^{2} b + a b^{2} - a b^{2} - b^{3}

SimpTerms

Simplify terms

a^{3} - b^{3}

Thus, the identity

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

is true.

Solve the equation

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Exercise

Find all complex solutions to the equation $x^{3} - 4^{3} = 0$ by using the difference of two cubes.

Show Solution

Solution

To solve the equation

x^{3} - 4^{3} = 0,

notice that the expression on the left-hand side is the difference of two cubes. We can use that to rewrite it as a product.

x^{3} - 4^{3} = 0 ⇕ (x - 4) (x^{2} + x \cdot 4 + 4^{2}) = 0

The Zero Product Property tells us that we can solve the equation by setting each of the factors equal to

0

and solving the resulting equations separately.

x - 4 = 0 and x^{2} + x \cdot 4 + 4^{2} = 0

Let's find the solution to the first equation.

x - 4 = 0 \Leftrightarrow x = 4

The second equation we will solve using the Quadratic Formula.

x^{2} + x \cdot 4 + 4^{2} = 0

CommutativePropMult

Commutative Property of Multiplication

x^{2} + 4 x + 4^{2} = 0

CalcPow

Calculate power

x^{2} + 4 x + 16 = 0

UseQuadForm

Use the Quadratic Formula: $a = 1, b = 4, c = 16$

x = \frac{- 4 \pm 4 ^{2} - 4 \cdot 1 \cdot 1 6}{2 \cdot 1}

CalcPowProd

Calculate power and product

x = \frac{- 4 \pm 1 6 - 6 4}{2}

SubTerm

Subtract term

x = \frac{- 4 \pm - 4 8}{2}

SqrtNegToISqrt

$- a = i a$

x = \frac{- 4 \pm i 4 8}{2}

StateSol

State solutions

x = \frac{- 4 + i 4 8}{2} x = \frac{- 4 - i 4 8}{2}

We should simplify these expressions, if possible. The real part becomes

\frac{- 4}{2} = - 2

and the imaginary part can be simplified.

\frac{i 4 8}{2}

SplitIntoFactors

Split into factors

\frac{i 1 2 \cdot 4}{2}

SqrtProd

$a \cdot b = a \cdot b$

\frac{i 1 2 \cdot 4}{2}

CalcRoot

Calculate root

\frac{i 1 2 \cdot 2}{2}

SimpQuot

Simplify quotient

i 12

After we have simplified, we the complex solutions to the quadratic equation can be written as

x = - 2 + i 12

and

x = - 2 - i 12 .

x = 4, x = - 2 + i 12 and x = - 2 - i 12

Rule

Square of a Sum of Two Squares

Rule

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

To prove the identity

(a^{2} + b^{2})^{2} = (a^{2} - b^{2})^{2} + (2 a b)^{2},

it is enough to show that the right-hand side can be rewritten to equal the expression on the left-hand side.

(a^{2} - b^{2})^{2} + (2 a b)^{2}

ExpandNegPerfectSquare

$(a - b)^{2} = a^{2} - 2 a b + b^{2}$

(a^{2})^{2} - 2 a^{2} b^{2} + (b^{2})^{2} + (2 a b)^{2}

PowProd

$(a \cdot b)^{m} = a^{m} \cdot b^{m}$

(a^{2})^{2} - 2 a^{2} b^{2} + (b^{2})^{2} + 2^{2} a^{2} b^{2}

CalcPow

Calculate power

(a^{2})^{2} - 2 a^{2} b^{2} + (b^{2})^{2} + 4 a^{2} b^{2}

CommutativePropAdd

Commutative Property of Addition

(a^{2})^{2} + 4 a^{2} b^{2} - 2 a^{2} b^{2} + (b^{2})^{2}

SimpTerms

Simplify terms

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

The three terms can be factored, as they form a perfect square trinomial.

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

FacPosPerfectSquare

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

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

By that, the proof of the identity is complete.

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

Proving Polynomial Identities

Rule

Square of a Binomial

Rule

Rule

Example

Rewrite the expressions as perfect square trinomials

Rule

Product of a Conjugate Pair of Binomials

Rule

Rule

Sum of Two Cubes

Rule

Rule

Difference of Two Cubes

Rule

Example

Solve the equation

Rule

Square of a Sum of Two Squares

Rule