Given P(x)=2x2−x+2 and Q(x)=2x−1, what is P(Q(x)) ?

2 Answers
George C.
Nov 11, 2017

P(Q(x))=8x2−10x+5

Explanation:

Note that changing the variable name does not really affect what the polynomial is. So we can write:

P(t)=2t2−t+2

Then, substituting Q(x) for t it may be clear that:

P(Q(x))=2(Q(x))2−Q(x)+2

P(Q(x))=2(2x−1)2−(2x−1)+2

P(Q(x))=2(4x2−4x+1)−(2x−1)+2

P(Q(x))=(8x2−8x+2)−(2x−1)+2

P(Q(x))=8x2−10x+5

Y
Nov 11, 2017

Answer: P(Q(x))=8x2−10x+5

Explanation:

This is a composition of functions problem. Note that P(Q(x)) means to substitute Q(x) as the "x" variable in the P(x) expression.

Therefore, given P(x)=2x2−x+2 and Q(x)=2x−1:
P(Q(x))=P(2x−1)
=2(2x−1)2−(2x−1)+2

We can continue simplifying by noting that
(a±b)2=a2±2ab+b2

So:
2(2x−1)2−(2x−1)+2
=2(4x2−4x+1)−2x+1+2 by the distributive property
=8x2−8x+2−2x+3 by the distributive property again
=8x2−10x+5 which is our answer.

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