Brittney Lord

2021-02-08

Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same  Make an effort to mentally evaluate P(2) and Q(2) using the provided forms. What is simpler? Now write the polynomial $R\left(x\right)={x}^{5}—2{x}^{4}+3{x}^{3}—2{x}^{3}+3x+4$ in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head. Do you see how using the nested form has the same arithmetic steps as using synthetic division to determine a polynomial's value?

StrycharzT

Expert

Given

$P\left(x\right)=3{x}^{4}-5{x}^{3}+{x}^{2}-3x+5$

$Q\left(x\right)=\left(\left(\left(3x-5\right)x+1\right)x-3\right)x+5$

$R\left(x\right)={x}^{5}-2{x}^{4}+3{x}^{3}-2{x}^{2}+3x+4$

Expand Q

$Q\left(x\right)=\left(\left(\left(3x-5\right)x+1\right)x-3\right)x+5$

$=\left(\left(3{x}^{2}-5x+1\right)x-3\right)x+5$

$=\left(3{x}^{3}-5{x}^{2}+x-3\right)x+5$

$=3{x}^{4}-5{x}^{3}+{x}^{2}-3x+5$

So, $P\left(x\right)=Q\left(x\right)=3{x}^{4}-5{x}^{3}+{x}^{2}-3x+5$

Hence proved Evaluate P(2) and Q(2)

$P\left(x\right)=3{x}^{4}-5{x}^{3}+{x}^{2}-3x+5$

$P\left(2\right)=3{\left(2\right)}^{4}-5{\left(2\right)}^{3}+{\left(2\right)}^{2}-3\left(2\right)+5$

$=48-40+4-6+5$

$=11$

$Q\left(2\right)=\left(\left(\left(3\left(2\right)-5\right)2+1\right)2-3\right)2+5$

$=\left(\left(3\left(2\right)+1\right)2-3\right)2+5$

$=\left(\left(3\left(2\right)\right)-3\right)2+5$

$=\left(3\right)2+5$

$=11$
Nested form of R(x)

$R\left(x\right)={x}^{5}-2{x}^{4}+3{x}^{3}-2{x}^{2}+3x+4$

$R\left(x\right)=\left({x}^{4}-2{x}^{3}+3{x}^{2}-2x+3\right)x+4$

$=\left(\left({x}^{3}-2{x}^{2}+3x-2\right)x+3\right)x+4$

Jeffrey Jordon

Expert