Expand Q to prove that the polynomials P and Q are the same. P(x)=3x^{4}-5x^{3}+x^{2}-3x+5

rhita3yp

rhita3yp

Answered question

2022-03-01

Expand Q to prove that the polynomials P and Q are the same.
P(x)=3x45x3+x23x+5
Q(x)=(((3x5)x+1)x3)x+5
Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial R(x)=x52x4+3x32x2+3x+4 in "nested" form, like the polynomial Q. Use the nested form to find R(3) in your head.
Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value of a polynomial using synthetic division?

Answer & Explanation

flytandikqk

flytandikqk

Beginner2022-03-02Added 4 answers

Given:
P(x)=3x45x3+x23x+5
Q(x)=(((3x5)x+1)x3)x+5
R(x)=x52x4+3x32x2+3x+4
To find P(2):
P(x)=3x45x3+x23x+5
P(2)=3(2)45(2)3+(2)23(2)+5
=3(16)5(8)+46+5
=4840+46+5
=5746
=11
To find Q(2):
Q(x)=(((3x5)x+1)x3)x+5
Q(2)=(((3(2)5)2+1)23)2+5
=(((1)2+1)23)2+5
(63)2+5
3(2)+5
6+5
=11
Now we have to write R(x) in nested form.
R(x)=x52x4+3x32x2+3x+4
Now we have to factor out x.
R(x)=(x42x3+3x22x+3)x+4
=((x32x2+3x2)x+3)x+4
=(((x22x+3)x2)x+3)x+4
=((((x2)x+3)x2)x+3)x+4
copausc20

copausc20

Beginner2022-03-03Added 8 answers

Given polynomials are:
P(x)=3x45x3+x23x+5
Q(x)=(((3x5)x+1)x3)x+5
First we write nested form in simple polynomial:
Q(x)=(((3x5)x+1)x3)x+5

(((3x5)x+1)x23x)+5
((3x5)x(x23x)+1(x23x))+5
((3x5)x33x2+x23x)+5
(3x(x33x2+x23x)5(x33x2+x23x))+5
(3x49x3+3x39x25x3+15x25x2+15x)+5
3x411x3+x2+15x+5
Now, P(2)=3(2)45(2)+(2)23(2)+5
=3(16)5(8)+46+5
=4840+3
=5140
=11
Q(x)=3(2)411(2)3+(2)2+15(2)+5
=3(16)11(8)+4+30+5
=4888+39
=8788
=1
Given

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?