Back to Tutor Intake Assessment: Prove and use polynomial identities

HSA.APR.C.4 Tutor Intake — Polynomial Identities

This short check helps your tutor understand where to start with polynomial identities. Answer each question on your own without help. If you are unsure, give your best try.

Grade 9·5 problems·~10 min·Common Core Math - HS Algebra·standard·hsa-apr-c-4

Work through problems with immediate feedback

Concepts

Which of the following is a polynomial identity (true for all values
of $x$ and $y$ )?

$x^2 - 4 = 0$

$(x + y)^2 = x^2 + y^2$

$(x + y)(x - y) = x^2 - y^2$

$x^2 + y^2 = 0$

A student wants to prove the identity $(a + b)^2 = a^2 + 2ab + b^2$ .
Which approach correctly proves the identity?

Substitute $a = 1, b = 2$ : $(1+2)^2 = 9$ and $1 + 4 + 4 = 9$ .
Since both sides equal 9, the identity is proved.

Start from the left side: $(a + b)^2 = (a + b)(a + b)$
$= a^2 + ab + ba + b^2 = a^2 + 2ab + b^2$ . This equals the
right side.

Start from the right side: $a^2 + 2ab + b^2 = (a + b)(a + b) = (a + b)^2$ . This equals the left side.

Write $(a + b)^2 = a^2 + 2ab + b^2$ , then subtract $a^2 + b^2$
from both sides to get $2ab = 2ab$ , which is always true.

Procedures

Use the difference-of-squares identity to compute $97 \times 103$
without a calculator.

Enter the result.

Expand $(x + 5)^2$ using the perfect-square identity
$(a + b)^2 = a^2 + 2ab + b^2$ .

Enter the coefficient of $x$ in the expanded form.

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