1 Introduction to Factor Theorem
2 Finding missing coefficients in a polynomial given roots information ex1
3 Finding Polynomial given root information ex2
4 Remainder Theorem Challenging Ex
5 Remainder Theorem Challenging Ex2
Write the binomial factor that corresponds to the polynomial P(x).
P(4) = 0
Write the binomial factor that corresponds to the polynomial P(x)
.
P(-3) = 0
Write the binomial factor that corresponds to the polynomial P(x).
P(\frac{2}{3}) = 0
Write the binomial factor that corresponds to the polynomial P(x)
.
P(-\frac{1}{4}) = 0
Determine if x + 3
is a factor of \displaystyle
x^3 + x^2 - x + 6
Determine if x + 3
is a factor of \displaystyle
2x^3 + 9x^2 + 10x + 3
Determine if x + 3
is a factor of \displaystyle
x^3 +27
List the values that could be zeros of each polynomial. Then, factor the polynomial.
x^3 + 3x^2 -6x - 8
List the values that could be zeros of each polynomial. Then, factor the polynomial.
x^3 + 4x^2 -15x - 18
List the values that could be zeros of the polynomial. Then, factor the polynomial.
\displaystyle
x^3 -3x^2 -10x + 24
List the values that could be zeros of each polynomial. Then, factor the polynomial.
x^3 + x^2 -9x - 9
List the values that could be zeros of each polynomial. Then, factor the polynomial.
x^3 - x^2 -16x + 16
List the values that could be zeros of each polynomial. Then, factor the polynomial.
2x^3 - x^2 -72x + 36
Factor each polynomial by grouping terms.
\displaystyle
3x^3 +2x^2 -75x - 50
Factor each polynomial by grouping terms.
\displaystyle
3x^3 + 2x^2 -75x -50
Factor each polynomial by grouping terms.
\displaystyle
2x^4 + 3x^3 -32x^2 -48x
List the values that could be zeros of each polynomial. Then, factor the polynomial.
3x^3 - x^2 -72x + 36
List the values that could be zeros of each polynomial. Then, factor the polynomial.
2x^3 -9x^2 + 10x -3
Determine the values that could be zeros of the polynomial. Then, factor the polynomial.
\displaystyle
6x^3 -11x^2 -26x + 15
Determine the values that could be zeros of the polynomial. Then, factor the polynomial.
\displaystyle
4x^3 +3x^2 -4x - 3
Factor each polynomial.
x^3 +2x^2 - x -2
Factor each polynomial.
x^3 +4x^2 -7 x -10
Factor each polynomial.
x^3 +5x^2 -4 x+20
Factor each polynomial.
\displaystyle
x^3 + 5x^2 + 3x -4
Factor each polynomial.
\displaystyle
x^3 -4x^2 -11x + 30
Factor each polynomial.
x^4 - 4x^3 -x^2 + 16x - 12
Factor each polynomial.
8x^3 + 4x^2 -2x -1
Factor each polynomial.
\displaystyle
2x^3 + 5x^2 -x - 6
Factor each polynomial.
\displaystyle
5x^3 +3x^2 -x - 6
Factor each polynomial.
\displaystyle
6x^4 + x^3 -8x^2 - x + 2
Factor each polynomial.
\displaystyle
5x^4 +x^3 -22x^2 -4x + 8
Factor each polynomial.
\displaystyle
5x^4 +x^3 -22x^2 -4x + 8
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
.
Determine the value of k so that x + 2
is a factor of x^3 -2kx^2 + 6x -4
.
Determine the value of k
so that 3x - 2
is a factor of 3x^3 - 5x^2 + kx + 2
.
Factor each polynomial.
\displaystyle
2x^3 + 5x^2 -x - 6
Factor each polynomial.
\displaystyle
2x^3 + 5x^2 -x - 6
Factor each polynomial.
\displaystyle
6x^3+5x^2-21x + 10
Factor each polynomial.
\displaystyle
4x^3 -8x^2 + 3x - 6
Factor each polynomial.
\displaystyle
2x^3 + x^2 + x - 1
Factor each polynomial.
\displaystyle
x^4 -15x^2 - 10x + 24
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
Factor \displaystyle
x^3 -a^3
Factor
\displaystyle
x^3 -125
Factor
\displaystyle
8x^3 -1
125x^6 - 8
64x^{12} -27
\frac{8}{125}x^{3} -64y^6
a) Factor each sum of cubes.
\displaystyle
x^3 +1
x^3 + 8
64x^{12}-27
\frac{8}{125}x^{3}-64y^6
b) Factor \displaystyle
x^3 + a^3
Factor \displaystyle
x^3 +125
Factor
\displaystyle
8x^3 +1
125x^6 + 8
64x^{12} +27
\frac{8}{125}x^{3} +64y^6
Show that the polynomial x^4 + x^2 + 1
is not factorable into linear factors with integer coefficients.
Factor by letting m = x^2
.
\displaystyle
4x^4 -37x^2 + 9
Factor by letting m = x^2
.
\displaystyle
9x^4 -148x^2 + 64
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)
.
Factor each polynomial.
\displaystyle
2x^5 + 3x^4 -10x^3 -15x^2 + 8x + 12
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
.
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
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
Factor each expression.
\displaystyle
x^5 -1
Factor each expression.
\displaystyle
x^4 -1
\displaystyle
x^4 -16
\displaystyle
x^5 -1
\displaystyle
x^5 -32
Factor each expression.
\displaystyle
x^4 -16
Factor each expression.
\displaystyle
x^5 -32
Factor \displaystyle
x^n -a^n
Factor \displaystyle
x^6- 1
Factor
\displaystyle
x^5 - 243
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)
.