How does one write the polynomial p(x)=12x3+(−32)x2+1 using the standard basis [1,x,x2,x3] ?

It writes itself! Specifically, you want to represent p(x) as a linear combination of the polynomials 1,x,x2 and x3, which simply means that you want to write it in the formc0⋅1+c1x+c2x2+c3x3for some scalars c0,c1,c2 and c3. And the polynomial p(x) itself does just that:p(x)=1⋅1+0⋅x+(−32)x2+12x3with coefficients 1,0,−32, and 12.

Expert Answer to "How does one write the polynomial p(x)=12x3+(−32)x2+1 using..."

High School
Calculus and Analysis
Precalculus
Polynomials

hadaasyj

Answered question

2022-04-23

How does one write the polynomial

p (x) = \frac{1}{2} x^{3} + (- \frac{3}{2}) x^{2} + 1

using the standard basis

[1, x, x^{2}, x^{3}]

Answer & Explanation

tswe0uk

Beginner2022-04-24Added 19 answers

It writes itself! Specifically, you want to represent

p (x)

as a linear combination of the polynomials

1, x, x^{2}

and

x^{3}

, which simply means that you want to write it in the form

c_{0} \cdot 1 + c_{1} x + c_{2} x^{2} + c_{3} x^{3}

for some scalars

c_{0}, c_{1}, c_{2}

and

c_{3}

. And the polynomial p(x) itself does just that:

p (x) = 1 \cdot 1 + 0 \cdot x + (- \frac{3}{2}) x^{2} + \frac{1}{2} x^{3}

with coefficients

1, 0, - \frac{3}{2}

, and

\frac{1}{2}

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