2.2 Factor Theorem
Chapter
Chapter 2
Section
2.2
Lectures 6 Videos
Solutions 67 Videos

Write the binomial factor that corresponds to the polynomial P(x).

P(4) = 0

Q1a

Write the binomial factor that corresponds to the polynomial P(x).

P(-3) = 0

Q1b

Write the binomial factor that corresponds to the polynomial P(x).

P(\frac{2}{3}) = 0

Q1c

Write the binomial factor that corresponds to the polynomial P(x).

P(-\frac{1}{4}) = 0

Q1d

Determine if x + 3 is a factor of \displaystyle x^3 + x^2 - x + 6 

Q2a

Determine if x + 3 is a factor of \displaystyle 2x^3 + 9x^2 + 10x + 3 

Q2b

Determine if x + 3 is a factor of \displaystyle x^3 +27 

Q2c

List the values that could be zeros of each polynomial. Then, factor the polynomial.

x^3 + 3x^2 -6x - 8

Q3a

List the values that could be zeros of each polynomial. Then, factor the polynomial.

x^3 + 4x^2 -15x - 18

Q3b

List the values that could be zeros of the polynomial. Then, factor the polynomial.

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

Q3c

List the values that could be zeros of each polynomial. Then, factor the polynomial.

x^3 + x^2 -9x - 9

Q4a

List the values that could be zeros of each polynomial. Then, factor the polynomial.

x^3 - x^2 -16x + 16

Q4b

List the values that could be zeros of each polynomial. Then, factor the polynomial.

2x^3 - x^2 -72x + 36

Q4c

Factor each polynomial by grouping terms.

\displaystyle 3x^3 +2x^2 -75x - 50 

Q4d

Factor each polynomial by grouping terms.

\displaystyle 3x^3 + 2x^2 -75x -50 

Q4e

Factor each polynomial by grouping terms.

\displaystyle 2x^4 + 3x^3 -32x^2 -48x 

Q4f

List the values that could be zeros of each polynomial. Then, factor the polynomial.

3x^3 - x^2 -72x + 36

Q5a

List the values that could be zeros of each polynomial. Then, factor the polynomial.

2x^3 -9x^2 + 10x -3

Q5b

Determine the values that could be zeros of the polynomial. Then, factor the polynomial.

\displaystyle 6x^3 -11x^2 -26x + 15 

Q5c

Determine the values that could be zeros of the polynomial. Then, factor the polynomial.

\displaystyle 4x^3 +3x^2 -4x - 3 

Q5d

Factor each polynomial.

x^3 +2x^2 - x -2

Q6a

Factor each polynomial.

x^3 +4x^2 -7 x -10 

Q6b

Factor each polynomial.

x^3 +5x^2 -4 x+20 

Q6c

Factor each polynomial.

\displaystyle x^3 + 5x^2 + 3x -4 

Q6d

Factor each polynomial.

\displaystyle x^3 -4x^2 -11x + 30 

Q6e

Factor each polynomial.

x^4 - 4x^3 -x^2 + 16x - 12 

Q6f

Factor each polynomial.

8x^3 + 4x^2 -2x -1 

Q7a

Factor each polynomial.

\displaystyle 2x^3 + 5x^2 -x - 6 

Q7b

Factor each polynomial.

\displaystyle 5x^3 +3x^2 -x - 6 

Q7c

Factor each polynomial.

\displaystyle 6x^4 + x^3 -8x^2 - x + 2 

Q7d

Factor each polynomial.

\displaystyle 5x^4 +x^3 -22x^2 -4x + 8 

Q7e

Factor each polynomial.

\displaystyle 5x^4 +x^3 -22x^2 -4x + 8 

Q7f

An artist creates a carving from a rectangular block of soapstone whose volume, V, in cubic metres, can be modelled by V(x) = 6x^3 + 25x2^ + 2x - 8. Determine possible dimensions of the block, in metres, in terms of binomials of x.

Q8

Determine the value of k so that x + 2 is a factor of x^3 -2kx^2 + 6x -4.

Q9

Determine the value of k so that 3x - 2 is a factor of 3x^3 - 5x^2 + kx + 2.

Q10

Factor each polynomial.

\displaystyle 2x^3 + 5x^2 -x - 6 

Q11a

Factor each polynomial.

\displaystyle 2x^3 + 5x^2 -x - 6 

Q11b

Factor each polynomial.

\displaystyle 6x^3+5x^2-21x + 10 

Q11c

Factor each polynomial.

\displaystyle 4x^3 -8x^2 + 3x - 6 

Q11d

Factor each polynomial.

\displaystyle 2x^3 + x^2 + x - 1 

Q11e

Factor each polynomial.

\displaystyle x^4 -15x^2 - 10x + 24 

Q11f

Factor each difference of cubes.

i) \displaystyle x^3 -1 

ii) \displaystyle x^3 -8 

iii) \displaystyle x^3 -27 

iv) \displaystyle x^3 -64 

Q12a

Factor \displaystyle x^3 -a^3 

Q12b

Factor \displaystyle x^3 -125 

Q12c

Factor

• i) \displaystyle 8x^3 -1 
• ii) 125x^6 - 8
• iii) 64x^{12} -27
• iv) \frac{8}{125}x^{3} -64y^6
Q12d

a) Factor each sum of cubes.

• i) \displaystyle x^3 +1 
• ii) x^3 + 8
• iii) 64x^{12}-27
• iv) \frac{8}{125}x^{3}-64y^6

b) Factor \displaystyle x^3 + a^3 

Q13ab

Factor \displaystyle x^3 +125 

Q13c

Factor

• i) \displaystyle 8x^3 +1 
• ii) 125x^6 + 8
• iii) 64x^{12} +27
• iv) \frac{8}{125}x^{3} +64y^6
Q13d

Show that the polynomial x^4 + x^2 + 1 is not factorable into linear factors with integer coefficients.

Q14

Factor by letting m = x^2.

\displaystyle 4x^4 -37x^2 + 9 

Q15a

Factor by letting m = x^2.

\displaystyle 9x^4 -148x^2 + 64 

Q15b

Best of U has produced a new body wash. The profit, P, in dollars, can be modelled by the function P(x) x^3 - 6x^2 + 9x, where x is the number of bottles sold, in thousands.

a) Use the factor theorem to determine if x-1 is a factor of P(x).

Q16a

Factor each polynomial.

\displaystyle 2x^5 + 3x^4 -10x^3 -15x^2 + 8x + 12 

Q17a

Determine the values of m and n so that the polynomials 2x^3 + mx^2 + nx -3 and x^3 -3mx^2 + 2nx + 4 are both divisible by x -2.

Q18

Determine a polynomial function P(x) that satisfies each set of conditions.

P(-4) = P(-\frac{3}{4}) = P(\frac{1}{2}) = 0 and P(-2) = 50

Q19a

Determine a polynomial function P(x) that satisfies each set of conditions.

P(3) = P(-1) = P(\frac{1}{2}) = P( -\frac{3}{2}) = 0 and P(1) = -18

Q19b

Factor each expression.

\displaystyle x^5 -1 

Q20a

Factor each expression.

• i) \displaystyle x^4 -1 
• ii) \displaystyle x^4 -16 
• iii) \displaystyle x^5 -1 
• iv) \displaystyle x^5 -32 
Q20a

Factor each expression.

\displaystyle x^4 -16 

Q20a

Factor each expression.

\displaystyle x^5 -32 

Q20a

Factor \displaystyle x^n -a^n 

Q20b

Factor \displaystyle x^6- 1 

Q20c

Factor

\displaystyle x^5 - 243 

Q20d

Factor

\displaystyle x^4 -625 

Is there a pattern for factoring x^n + a^n? Justify your answer.
When a polynomial is divided by (x + 2), the remainder is -19. When the same polynomial is divided by (x - 1), the remainder is 2. Determine the remainder when the polynomial is divided by (x - 1)(x + 2).