Task Sets: Prove and use polynomial identities - Common Core Math - HS Algebra

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Which statement correctly distinguishes a polynomial identity from a polynomial equation?

A polynomial identity is true for specific values of the variable; an equation is true for all values.

A polynomial identity is true for all values of the variable; an equation is true only for specific values.

Both identities and equations must be solved to find the values that make them true.

A polynomial identity always uses the symbol $\equiv$ ; a polynomial equation always uses $=$ .

Which of these is a polynomial identity (true for all values of $x$ )?

$x^2 - 4 = 0$

$(x + 2)(x - 2) = x^2 - 4$

$x^2 = 4$

$x + 2 = 6$

Which expression correctly states the difference of squares identity?

$a^2 - b^2 = (a - b)^2$

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

$a^2 - b^2 = (a - b)(a - b)$

$a^2 - b^2 = a^2 - 2ab + b^2$

A student wants to prove that $(x + y)^2 = x^2 + 2xy + y^2$ is a polynomial identity. Which approach is mathematically valid?

Test 5 different values of $x$ and $y$ , confirm the equation holds for each, and conclude it is an identity.

Expand $(x + y)(x + y)$ algebraically to get $x^2 + 2xy + y^2$ , confirming the two sides are equal as expressions.

Set the two sides equal and solve for $x$ to find all values where it holds.

Verify with $x = 0$ and $y = 0$ only, since zero is the simplest test.

$Two-column proof scaffold showing correct left-to-right transformation of (a−b)² into a²−2ab+b², with an arrow showing it matches the right side.$

To prove the identity $(a - b)^2 = a^2 - 2ab + b^2$ , which sequence of steps is correct?

Start: $(a-b)^2 = a^2 - 2ab + b^2$ . Square-root both sides. Done.

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

Start from both sides simultaneously, add $2ab$ to both sides, and conclude $a^2 = (a-b)^2 + 2ab - b^2$ .

Substitute $a = 1, b = 0$ and verify: $(1-0)^2 = 1$ and $1 - 0 + 0 = 1$ . Identity proved.

Which is the correct factorization for the difference of cubes $a^3 - b^3$ ?

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

$a^3 - b^3 = (a + b)(a^2 - ab + b^2)$

$a^3 - b^3 = (a - b)(a^2 - ab - b^2)$

$a^3 - b^3 = (a - b)^3$

Use the difference of squares identity $a^2 - b^2 = (a+b)(a-b)$ to compute $99 \times 101$ mentally.

Enter the value of $99 \times 101$ .

Use the perfect square identity $(a + b)^2 = a^2 + 2ab + b^2$ to compute $52^2$ mentally.

Write $52 = 50 + 2$ , then apply the identity. Enter the value of $52^2$ .

Use the Pythagorean-triple identity with $x = 2$ , $y = 1$ to find the value of $c = x^2 + y^2$ .

Enter the value of $c$ .

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