High School

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When a polynomial [tex]\varnothing(x) = dx^3 + fx^2 - 7x - 6[/tex] is divided by [tex]x - 1[/tex], the remainder is -10. When it is divided by [tex]x - 3[/tex], the remainder is 36.

Find the values of [tex]d[/tex] and [tex]f[/tex].

Answer :

To find the values of [tex]\( d \)[/tex] and [tex]\( f \)[/tex] for the polynomial [tex]\( \varphi(x) = d x^3 + f x^2 - 7 x - 6 \)[/tex], we use the Remainder Theorem. According to this theorem, the remainder of a polynomial [tex]\( \varphi(x) \)[/tex] when divided by [tex]\( x-a \)[/tex] is equal to [tex]\( \varphi(a) \)[/tex].

Let's solve this step-by-step:

1. Given Conditions:
- When the polynomial is divided by [tex]\( x-1 \)[/tex], the remainder is [tex]\(-10\)[/tex]. Thus, [tex]\(\varphi(1) = -10\)[/tex].
- When the polynomial is divided by [tex]\( x-3 \)[/tex], the remainder is [tex]\(36\)[/tex]. Thus, [tex]\(\varphi(3) = 36\)[/tex].

2. Set Up the Equations:

Substitute [tex]\( x = 1 \)[/tex] into the polynomial:
[tex]\[
\varphi(1) = d(1)^3 + f(1)^2 - 7(1) - 6 = -10
\][/tex]
Simplifying, we get:
[tex]\[
d + f - 7 - 6 = -10 \\
d + f - 13 = -10 \\
d + f = 3 \quad \text{(Equation 1)}
\][/tex]

Substitute [tex]\( x = 3 \)[/tex] into the polynomial:
[tex]\[
\varphi(3) = d(3)^3 + f(3)^2 - 7(3) - 6 = 36
\][/tex]
Simplifying, we get:
[tex]\[
27d + 9f - 21 - 6 = 36 \\
27d + 9f - 27 = 36 \\
27d + 9f = 63 \quad \text{(Equation 2)}
\][/tex]

3. Solve the System of Equations:
We have two equations:
1. [tex]\( d + f = 3 \)[/tex]
2. [tex]\( 27d + 9f = 63 \)[/tex]

We can simplify Equation 2 by dividing all terms by 9:
[tex]\[
3d + f = 7
\][/tex]

Now we solve this system of equations:
- [tex]\( d + f = 3 \)[/tex]
- [tex]\( 3d + f = 7 \)[/tex]

Subtract the first equation from the second:
[tex]\[
(3d + f) - (d + f) = 7 - 3 \\
2d = 4 \\
d = 2
\][/tex]

Substitute [tex]\( d = 2 \)[/tex] back into [tex]\( d + f = 3 \)[/tex]:
[tex]\[
2 + f = 3 \\
f = 1
\][/tex]

Thus, the values are [tex]\( d = 2 \)[/tex] and [tex]\( f = 1 \)[/tex].

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