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Math  /  Algebra

QuestionWhich expression is equivalent to (z+5)(z2−4z+6)?(z+5)\left(z^{2}-4 z+6\right) ? (A) z3+5z2+6z+30z^{3}+5 z^{2}+6 z+30 (B) z2+z+11z^{2}+z+11 (C) z3+z2−14z+30z^{3}+z^{2}-14 z+30 (D) z2−3z+11z^{2}-3 z+11

Studdy Solution

STEP 1

1. We are given an expression (z+5)(z2−4z+6)(z+5)(z^2-4z+6) and need to find an equivalent expression.
2. The problem involves polynomial multiplication (specifically, distributing a binomial over a trinomial).

STEP 2

1. Distribute zz across the trinomial (z2−4z+6)(z^2 - 4z + 6).
2. Distribute 55 across the trinomial (z2−4z+6)(z^2 - 4z + 6).
3. Combine like terms to simplify the expression.
4. Match the simplified expression with the given options.

STEP 3

Distribute zz across the trinomial (z2−4z+6)(z^2 - 4z + 6):
z⋅(z2−4z+6)=z⋅z2+z⋅(−4z)+z⋅6 z \cdot (z^2 - 4z + 6) = z \cdot z^2 + z \cdot (-4z) + z \cdot 6
=z3−4z2+6z = z^3 - 4z^2 + 6z

STEP 4

Distribute 55 across the trinomial (z2−4z+6)(z^2 - 4z + 6):
5⋅(z2−4z+6)=5⋅z2+5⋅(−4z)+5⋅6 5 \cdot (z^2 - 4z + 6) = 5 \cdot z^2 + 5 \cdot (-4z) + 5 \cdot 6
=5z2−20z+30 = 5z^2 - 20z + 30

STEP 5

Combine the results from Step 1 and Step 2:
z3−4z2+6z+5z2−20z+30 z^3 - 4z^2 + 6z + 5z^2 - 20z + 30
Group like terms:
z3+(−4z2+5z2)+(6z−20z)+30 z^3 + (-4z^2 + 5z^2) + (6z - 20z) + 30
Simplify:
z3+z2−14z+30 z^3 + z^2 - 14z + 30

STEP 6

Match the simplified expression z3+z2−14z+30 z^3 + z^2 - 14z + 30 with the given options:
The expression matches option (C) z3+z2−14z+30 z^3 + z^2 - 14z + 30 .
The equivalent expression is C\boxed{C}.

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