3.6 Factoring Polynomials
Chapter
Chapter 3
Section
3.6
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Lectures 7 Videos

Intro IVT Intermediate Value Theorem and ex

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Intro IVT Intermediate Value Theorem and ex

example Find the value of k if the remainder of P(x) = 3x^2 -2kx + 5 when divided by x - 2 is 4.

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1.04mins
ex3 Finding k when remainder is given

example Find the value of a and b when P(x) =ax^3 -(b -1)x^2 + x - 2 is divisible by x^2 -2x - 3.

Key Ideas

x^2 -2x - 3 has two factors (x -3) and (x + 1).

Use Remainder theorem with the two factors.

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Using Factor Theorem to find the missing coefficients in the polynomial
Solutions 40 Videos

Given f(x) = x^4 + 5x^3 + 3x^2 - 7x + 10, determine the remainder when f(x) is divided by each of the following binomials, without dividing:

(i) x - 2

(ii) x + 4

(iii) x-1

Are any of the binomials in part (a) factors of f(x)? Explain.

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Q1

Which of the following functions are divisible by x- 1?

f(x) = x^4 - 15x^3 + 2x^2 + 12x - 10

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0.41mins
Q2a

Which of the following functions are divisible by x- 1?

f(x) = 5x^3 - 4x^2 + 3x -4

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Q2b

Is the function divisible by x- 1?

h(x) = x^4 - 7x^3 + 2x^2 + 9x

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Q2c

Is the functions are divisible by x- 1?

j(x) = x^3-1

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Q2d

Determine all the factors of the function f(x) = x^3 + 2x^2 - 5x - 6

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1.17mins
Q3

State the remainder when x + 2 is divided into polynomial x^2 + 7x + 9

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0.22mins
Q4a

State the remainder when x + 2 is divided into polynomial 6x^3 + 19x^2 + 11x -11

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Q4b

State the remainder when x + 2 is divided into polynomial x^4 - 5x^2 + 4

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0.26mins
Q4c

State the remainder when x + 2 is divided into polynomial x^4 - 2x^3 -11x^2 +10x -2

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Q4d

State the remainder when x + 2 is divided into polynomial x^3 + 3x^2 -10x +6

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0.28mins
Q4e

State the remainder when x + 2 is divided into polynomial 4x^4 +12x^3 - 13x^2 -33x + 18

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1.02mins
Q4f

Determine whether 2x - 5 is a factor of polynomial

x^3 - 3x^2 -10x + 24

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Q5a

Determine whether 2x - 5 is a factor of polynomial

4x^3 + 12x^2 -x - 15

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Q5b

Determine whether 2x - 5 is a factor of polynomial

x^4 + 8x^3 + 4x^2 -48x

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Q5c

Determine whether 2x - 5 is a factor of polynomial

4x^4 + 7x^3 - 80x^2 - 21x + 270

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2.02mins
Q5d

Factor each polynomial using the factor theorem.

x^3 - 3x^2 -10x + 24

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1.04mins
Q6a

Factor each polynomial using the factor theorem.

4x^3 + 12x^2 -x - 15

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1.08mins
Q6b

Factor each polynomial using the factor theorem.

x^4 + 8x^3 + 4x^2 -48x

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1.01mins
Q6c

Factor each polynomial using the factor theorem.

4x^4 + 7x^3 - 80x^2 - 21x + 270

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2.09mins
Q6d

Factor each polynomial using the factor theorem.

x^5 - 5x^4 - 7x^3 + 29x^2 + 30x

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2.31mins
Q6e

Factor each polynomial using the factor theorem.

x^4 + 2x^3 -23x^2 -24x + 144

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Q6f

Factor fully.

f(x) =x^3 + 9x^2 + 8x - 60

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Q7a

Factor fully.

f(x) = x^3 - 7x - 6

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Q7b

Factor fully.

f(x) = x^4 - 5x^2 + 4

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Q7c

Factor fully.

f(x) = x^4 + 3x^3-38x^2 +24x + 64

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Q7d

Factor fully.

f(x) =x^3 -x^2 + x -1

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Q7e

Factor fully.

f(x) = x^5 -x^4 + 2x^3 -2x^2 + x -1

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Q7f

Which of the following can be the graph of below?

f(x) = x^4 - 5x^2 + 4

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Q8c

Which of the following can be the graph of below?

f(x) = x^4 + 3x^3-38x^2 +24x + 64

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Q8d

Use the factored form of f(x) to sketch the graph of each functions.

f(x) =x^3 -x^2 + x -1

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Q8e

Use the factored form of f(x) to sketch the graph of each functions.

f(x) = x^5 -x^4 + 2x^3 -2x^2 + x -1

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Q8f

The polynomial 12x^3 + kx^2 - x - 6 has 2x - 1 as one of its factors. Determine the value of k.

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Q9

When ax^3 -x^2+2x + b is divided by x -1, the remainder is 10. When it is divided by x - 2, the remainder is 51. Find a and b.

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Q10

Determine a general rule to help decide whether x- a and x - a are factors of x^n -a^n and x^n + a^n.

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Q11

The function f(x) = ax^3 - x^2 + bx - 24 has three factors. Two of these factors are x - 2and x + 4. Determine the values of a and b, and then determine the other factor.

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Q12

Consider the function f(x) = x^3 + 4x^2 + kx - 4. The remainder from f(x) \div (x + 2) is twice the remainder from f(x) \div (x + 2). Determine the value of k.

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Q13

Show that x -a is a factor of x^4 - a^4.

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Q14

Use the factor theorem to prove that x^2 -x - 2 is a factor of x^3 - 6x^2 + 3x + 10.

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1.27mins
Q16

Prove that x + a is a factor of (x + a)^5 + (x + c)^5 + (a-c)^5.

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Q17