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Solutions
67 Videos

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

`P(4) = 0`

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Q1a

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

.

`P(-3) = 0`

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Q1b

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

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

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Q1c

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

.

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

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Q1d

Determine if `x + 3`

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

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Q2a

Determine if `x + 3`

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

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Q2b

Determine if `x + 3`

is a factor of ```
\displaystyle
x^3 +27
```

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Q2c

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

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

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Q3a

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

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

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Q3b

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

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

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Q3c

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

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

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Q4a

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

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

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Q4b

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

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

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Q4c

Factor each polynomial by grouping terms.

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

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Q4d

Factor each polynomial by grouping terms.

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

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Q4e

Factor each polynomial by grouping terms.

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

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Q4f

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

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

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Q5a

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

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

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Q5b

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

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

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Q5c

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

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

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Q5d

Factor each polynomial.

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

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Q6a

Factor each polynomial.

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

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Q6b

Factor each polynomial.

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

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Q6c

Factor each polynomial.

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

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Q6d

Factor each polynomial.

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

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Q6e

Factor each polynomial.

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

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Q6f

Factor each polynomial.

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

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Q7a

Factor each polynomial.

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

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Q7b

Factor each polynomial.

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

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Q7c

Factor each polynomial.

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

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Q7d

Factor each polynomial.

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

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Q7e

Factor each polynomial.

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

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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`

.

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Q8

Determine the value of k so that `x + 2`

is a factor of `x^3 -2kx^2 + 6x -4`

.

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Q9

Determine the value of `k`

so that `3x - 2`

is a factor of `3x^3 - 5x^2 + kx + 2`

.

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Q10

Factor each polynomial.

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

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Q11a

Factor each polynomial.

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

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Q11b

Factor each polynomial.

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

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Q11c

Factor each polynomial.

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

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Q11d

Factor each polynomial.

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

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Q11e

Factor each polynomial.

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

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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
```

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Q12a

Factor ```
\displaystyle
x^3 -a^3
```

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Q12b

Factor
```
\displaystyle
x^3 -125
```

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Q12c

Factor

- i)
`\displaystyle 8x^3 -1`

- ii)
`125x^6 - 8`

- iii)
`64x^{12} -27`

- iv)
`\frac{8}{125}x^{3} -64y^6`

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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
```

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Q13ab

Factor ```
\displaystyle
x^3 +125
```

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Q13c

Factor

- i)
`\displaystyle 8x^3 +1`

- ii)
`125x^6 + 8`

- iii)
`64x^{12} +27`

- iv)
`\frac{8}{125}x^{3} +64y^6`

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Q13d

Show that the polynomial `x^4 + x^2 + 1`

is not factorable into linear factors with integer coefficients.

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Q14

Factor by letting `m = x^2`

.

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

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Q15a

Factor by letting `m = x^2`

.

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

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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)`

.

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Q16a

Factor each polynomial.

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

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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`

.

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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`

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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`

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Q19b

Factor each expression.

```
\displaystyle
x^4 -16
```

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Q20a

Factor each expression.

- i)
`\displaystyle x^4 -1`

- ii)
`\displaystyle x^4 -16`

- iii)
`\displaystyle x^5 -1`

- iv)
`\displaystyle x^5 -32`

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Q20a

Factor each expression.

```
\displaystyle
x^5 -1
```

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Q20a

Factor each expression.

```
\displaystyle
x^5 -32
```

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Q20a

Factor ```
\displaystyle
x^n -a^n
```

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Q20b

Factor ```
\displaystyle
x^6- 1
```

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Q20c

Factor

```
\displaystyle
x^4 -625
```

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Q20d

Factor

```
\displaystyle
x^5 - 243
```

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Q20d

Is there a pattern for factoring `x^n + a^n`

? Justify your answer.

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Q21

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)`

.

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Q22

Lectures
4 Videos

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1.31mins

Introduction to Factor Theorem

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5.52mins

Finding missing coefficients in a polynomial given roots information ex1

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2.12mins

Finding Polynomial given root information ex2

5 Remainder Theorem Challenging Ex2

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6.18mins

Remainder Theorem Challenging Ex2