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Polynomial Functions

Proving Polynomial Identities

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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.

For simplicity, depending on the sign of the binomial, these two identities can be expressed as one.

(a±b)2=a2±2ab+b2

Proof

This rule will be first proven for (a+b)2 and then for (ab)2.

This identity can be shown by first rewriting the square as a product.
(a+b)2
(a+b)(a+b)
Multiply parentheses
a(a+b)+b(a+b)
a2+ab+b(a+b)
a2+ab+ba+b2
a2+ab+ab+b2
a2+2ab+b2
It has been shown that (a+b)2=a2+2ab+b2.

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
(ab)(ab)
Multiply parentheses
a(ab)b(ab)
a2abb(ab)
a2abba+b2
a2abab+b2
a22ab+b2
It has been shown that (ab)2=a22ab+b2.

Example

Rewrite the expressions as perfect square trinomials

fullscreen

Rewrite (x+3)2 and (7x)2 as perfect square trinomials.

Show Solution expand_more
When rewriting (x+3)2 as a perfect square trinomial, keep in mind that it is the square of a binomial.
(x+3)2
x2+2x3+32
x2+6x+32
x2+6x+9
The second expression has a minus sign between the terms – take this into account when rewriting the binomial.
(7x)2
7227x+x2
4927x+x2
4914x+x2
The expressions rewritten as perfect square trinomials are x2+6x+9 and 4914x+x2, respectively.

Rule

Product of a Conjugate Pair of Binomials

The product of two conjugate binomials is the difference of two squares.

(a+b)(ab)=a2b2

Proof

This identity can be proved by using the Distributive Property to multiply the binomials.
(a+b)(ab)
a(ab)+b(ab)
a2ab+b(ab)
a2ab+bab2
a2ab+abb2
a2b2
Therefore, the product of a binomial and its conjugate is the difference of two squares.

Rule

Sum of Two Cubes

Rule

To show the identity
it is enough to rewrite the right-hand side and show that it equals the sum on the left-hand side.
aa2aab+ab2+ba2bab+bb2
a3a2b+ab2+a2bab2+b3
a3a2b+a2b+ab2ab2+b3
a3+b3

Rule

Difference of Two Cubes

Rule

Showing this identity is easiest done by rewriting the right-hand side.
aa2+aab+ab2ba2babbb2
a3+a2b+ab2a2bab2b3
a3+a2ba2b+ab2ab2b3
a3b3
Thus, the identity is true.

Example

Solve the equation

fullscreen

Find all complex solutions to the equation x343=0 by using the difference of two cubes.

Show Solution expand_more
To solve the equation x343=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.
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.
Let's find the solution to the first equation.
The second equation we will solve using the Quadratic Formula.
x2+x4+42=0
x2+4x+42=0
x2+4x+16=0
We should simplify these expressions, if possible. The real part becomes and the imaginary part can be simplified.
After we have simplified, we the complex solutions to the quadratic equation can be written as and

Rule

Square of a Sum of Two Squares

Rule

To prove the identity
it is enough to show that the right-hand side can be rewritten to equal the expression on the left-hand side.
The three terms can be factored, as they form a perfect square trinomial.
By that, the proof of the identity is complete.


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