Welcome! Today we're tackling Factor Theorem Translations and Two-Unknown Problems — where you'll learn to handle problems that give you multiple pieces of information at once.

Some problems give you two pieces of information:

Clue Type	Example
Factor condition	"x minus 2 is a factor"
Relationship	"sum of zeros is 5"

...and ask you to find two unknownsgoal.

The strategy combines:

Substitution from the factor condition
Coefficient formula from the relationship

Two clues → Two equations → Two unknowns

In this lesson, you will:

Practise this combination strategy
Learn the remainder theorem extension

1. Combining factor condition with relationship formula

Let's see how you handle a problem that combines two different conditions about a polynomial.

When a problem gives you both a factor condition (like " $(x - 3)$ is a factor") and a zeros relationship (like "sum of zeros is 5"), you actually get two equations in two unknowns.

The factor condition gives one equation via substitution
The relationship formula gives the other equation

✍️ Question

Problem 📝

If $(x - 3)$ is a factor of $x^{3} + a x^{2} + b x - 12$ and the sum of zeros is 5, find $a$ and $b$ .

Given information:

$(x - 3)$ is a factor → $p(3) = 0$ condition 1
Sum of zeros is 5 → $\frac{-a}{1} = 5$ condition 2

Two pieces of information give two equationsThe pattern: two unknowns means find two conditions:

Equation 1 (sum formula)Sum of zeros equals negative x-squared coefficient over leading coefficient: For $x^{3} + a x^{2} + b x - 12$ , the sum of zeros $= \frac{-a}{1} = -a$ Negative a over 1 gives us negative a. Given sum $= 5$ : $- a = 5$ , so $a = -5$ found it!One condition, one unknown solved.

✍️ MCQ

Choose one

We used the sum of zeros formula to find

a = -5

. Which condition have we NOT used yet?

✓

Equation 2 (factor condition)Second condition from the factor: $(x - 3)$ is a factor means $p(3) = 0$ Translate 'is a factor' into p of that root equals zero. $p (3) = 27 + 9 a + 3 b - 12 = 0$

Substitute $a = -5$ into Equation 2:

27 + 9(-5) + 3b - 12 = 0

Simplify step by step:

27 - 45 + 3b - 12 = 0

-30 + 3b = 0

✍️ MCQ

Choose one

From

-30 + 3b = 0

, what is the value of

b

?

✓

Solve for $b$ :

$3 b = 30$

b = 10

(Using the Factor Theorem condition to find b)

The Strategy for Two-Unknown ProblemsYou need two equations to solve

When you have two equations with two unknowns ( $a$ and $b$ ), use this approach:

Start with the simpler equationIt gives you one variable directly — the sum of zeros formula gives us $a$ directly
Substitute into the harder equationUse the value you found — use the Factor Theorem equation to find $b$

Applying to our problem:

Simpler equation: Sum of zeros $= - \frac{a}{1} = 5$ , so $a = -5$ found directly
Harder equation: $p (3) = 0$ — substitute $a = -5$ here to find $b$ answer

✍️ MCQ

Choose one

In a two-unknown problem like this, why do we start with the sum of zeros equation rather than the Factor Theorem equation?

✓

2. Finding the other zero after determining the unknown

Once you find the unknown coefficient, the polynomial is fully determined.

You can then find all zeros using either method:

Sum formula — subtract the known zero from the total sum
Factoring — factor the polynomial directly

In our problem: once $k = 9$ found!, the polynomial becomes $x^{2} - x - 20$ , and finding the other zero is just one more step.

✍️ Question

Question 📝

If $-4$ is a zero of $x^2 - x - (2k + 2)$ , find $k$ and then find the other zero.

Step 1: Find k by substitution.

Since $-4$ is a zeroWhen a number is a zero, substituting it gives zero of the polynomial, we substitute $x = - 4$ :

p(-4) = (-4)^2 - (-4) - (2k + 2) = 0

(This is how we set up our equation to find k)

16 + 4 - 2k - 2 = 0

18 - 2k = 0

k = 9

Answer!

✍️ MCQ

Choose one

Now that we know

k = 9

, the polynomial becomes

x^2 - x - 20

. What should we do next to find the other zero?

✓

Step 2: With $k = 9$ , the polynomial becomes:

x^2 - x - (2(9) + 2) = x^2 - x - 20

(This is our actual polynomial after substituting k)

Now we need to find the other zeroA quadratic has two zeros and we only know one of this quadratic.

✍️ MCQ

Choose one

To find the other zero of

x^2 - x - 20

, which method would be most efficient here?

✓

Step 3: Find the other zero.

Method 1 (sum formula): For $x^{2} - x - 20$ , the sum of zeros $= -\frac{b}{a} = -\frac{-1}{1} = 1$ Sum formula: negative b divided by a.

If one zero is $- 4$ , the other is $1 - (-4) = 1 + 4 = 5$ Subtract known zero from sum.

✍️ FIB

Fill in the blank

If the sum of zeros of a quadratic is

7

and one zero is

3

, what is the other zero?

✓

Method 2 (factoring): For $x^{2} - x - 20$ , we need two numbers with product $a \cdot c = -20$ Product equals a times c and sum $= -1$ Sum comes from coefficient b. The pair is $-5$ and $4$ the pairThe pair that works.

Splitting: $(x - 5) (x + 4)$ . Zeros: $5$ and $- 4$ .

Both methods give the same answer. The sum formula is fasterQuicker when you have one zero for finding one more zero.