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Which of the following polynomials has a remainder of -7 when divided by x – 2?
A. 4x3 + 2x2 + 5
B. x3 - 2x2 - 4x + 1
c. 3x² + 6x - 2
D.-2x3 + 4x2 + 3x - 2

Respuesta :

MrRoyal

Answer:

[tex]B.\ x^3 - 2x^2 - 4x + 1[/tex]

Step-by-step explanation:

Given

Polynomials A to D

Divisor: x - 2

Required

Which of polynomial A - D has a remainder of -7

We start by equating the divisor to 0

[tex]x - 2 = 0[/tex]

Make x the subject of formula

[tex]x = 2[/tex]

Next is to substitute 2 for x in the polynomials to get the remainder;

[tex]A.\ 4x^3 + 2x^2 + 5[/tex]

[tex]4(2)^3 + 2(2)^2 + 5[/tex]

Open Brackets

[tex]4 * 8 + 2 * 4 + 5[/tex]

[tex]32 + 8 + 5[/tex]

[tex]Remainder = 45[/tex]

[tex]B.\ x^3 - 2x^2 - 4x + 1[/tex]

[tex](2)^3 - 2(2)^2 - 4(2) + 1[/tex]

Open Brackets

[tex]8 - 2 * 4 - 4 * 2 + 1[/tex]

[tex]8 - 8 - 8 + 1[/tex]

[tex]Remainder = -7[/tex]

[tex]C.\ 3x^2 + 6x - 2[/tex]

[tex]3(2)^2 + 6(2) - 2[/tex]

Open Brackets

[tex]3 * 4 + 6 * 2 - 2[/tex]

[tex]12 + 12 - 2[/tex]

[tex]Remainder = -2[/tex]

[tex]D.\ -2x^3 + 4x^2 + 3x - 2[/tex]

[tex]-2(2)^3 + 4(2)^2 + 3(2) - 2[/tex]

Open Brackets

[tex]-2 * 8 + 4 * 4 + 3 * 2 - 2[/tex]

[tex]-16 + 16 + 6 - 2[/tex]

[tex]Remainder = 4[/tex]

From the calculations above, the polynomial with a remainder of -7 when divided by [tex]x - 2[/tex] is [tex]B.\ x^3 - 2x^2 - 4x + 1[/tex]

The value of the polynomial [tex]x^3-2x^2-4x+1[/tex] is -7 at x = 2. So, option B is correct.

Important information:

  • Remainder is -7.
  • Divisor is [tex]x-2[/tex].

Remainder Theorem:

According to the Remainder Theorem, if a polynomial is p(x) is divided by (x-c), then the remainder is p(c).

If polynomials have a remainder of -7 when divided by x – 2. So, the value of the polynomial must be -7 at x = 2.

Substitute x = 2 in the first polynomial.

[tex]4(2)^3+2(2)^2+5=45\neq -7[/tex]

Substitute x = 2 in the second polynomial.

[tex](2)^3-2(2)^2-4(2)+1=-7[/tex]

Substitute x = 2 in the third polynomial.

[tex]3(2)^2+6(2)-2=22\neq -7[/tex]

Substitute x = 2 in the fourth polynomial.

[tex]-2(2)^3+4(2)^2+3(2)-2=4\neq -7[/tex]

Only the value of the second polynomial is -7 at x = 2. Therefore, the correct option is B.

Find out more about 'Remainder Theorem' here:

https://brainly.com/question/4515216

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