3.08 Proofs and further applications of polynomial identities
Many of the patterns used in factoring polynomials are mathematical identities.
While an equation might be true for some specific value of variables within it, a mathematical identity is true for every value of the variables within it.
Verifying an identity is considered a mathematical proof. We cannot manipulate both sides of the equation at once. Instead, we must work with one side of the equation and show that algebraic manipulation leads to the other side.
To demonstrate this, we'll work through verifying a few polynomial identities we've worked with before.
Worked examples
Question 1
Prove the identity $\left(a+b\right)^2=a^2+2ab+b^2$(a+b)2=a2+2ab+b2.
Think: We can prove the identity by showing through algebraic manipulation that the equation is true for all values of a and b.
Do: Let's start with the left-hand side of the equation and show that it leads to the right-hand side.
| $\left(a+b\right)^2$(a+b)2 | $=$= | $\left(a+b\right)^2$(a+b)2 |
The left-hand side of the equation |
| $=$= | $\left(a+b\right)\left(a+b\right)$(a+b)(a+b) |
Rewrite the power |
|
| $=$= | $a^2+ab+ab+b^2$a2+ab+ab+b2 |
Distribute the expression, multiplying all four terms |
|
| $=$= | $a^2+2ab+b^2$a2+2ab+b2 |
Simplify |
Therefore, we have proven the identity because the expressions on both sides of the equation are identical for all values of $a$a and $b$b.
Practice questions
Question 2
Prove that $\left(m-n\right)^2=m^2-2mn+n^2$(m−n)2=m2−2mn+n2, showing all steps of work.
Question 3
Consider the difference of two squares expression $u^2-v^2$u2−v2.
Express the difference of two squares as a product.
Use the identity $u^2-v^2=\left(u+v\right)\left(u-v\right)$u2−v2=(u+v)(u−v) to find the value of $77^2-23^2$772−232.
Use the identity $u^2-v^2=\left(u+v\right)\left(u-v\right)$u2−v2=(u+v)(u−v) to find the value of $24\times16$24×16.
Question 4
Answer the following.
Show that $\left(x-a\right)\left(x^2+ax+a^2\right)=x^3-a^3$(x−a)(x2+ax+a2)=x3−a3.
Hence show that $999973$999973 is not prime by expressing it as the product of two factors.