Polynomials · Learner pages

Lesson 1 · Zeroes and the x-axis

Connect a zero, an input value and an x-intercept.

Name or initials: ____________   Date: ____________

A polynomial gives an output for each input. A zero is an input whose output is zero.

1. Evaluate the input. For p(x) = (x − 1)(x − 3), substitute a candidate value for x.
2. Test x = 1. p(1) = (1 − 1)(1 − 3) = 0 × (−2) = 0.
3. Test x = 3. p(3) = (3 − 1)(3 − 3) = 2 × 0 = 0. The graph meets the x-axis at 1 and 3.
4. Separate input and point. The zeroes are the numbers 1 and 3. The corresponding points are (1, 0) and (3, 0). The sketch is schematic.

Worked example: For p(x) = x² − 4x + 3, p(1) = 1 − 4 + 3 = 0 and p(3) = 9 − 12 + 3 = 0. Thus 1 and 3 are zeroes; p(0) = 3 is the y-intercept.

1. For p(x) = (x − 1)(x − 3), which listed value is a zero?

   1. 1 · 2. 11 · 3. 21

2. For p(x) = (x − 2)(x − 4), which listed value is a zero?

   1. 12 · 2. 22 · 3. 2

3. For p(x) = (x − 3)(x − 5), which listed value is a zero?

   1. 23 · 2. 3 · 3. 13

Discuss: Choose (x − 2)(x − 5). Make a table for x = 0 through 6 and mark the two zero outputs.

Lesson 2 · Factor a quadratic to find its zeroes

Use multiplication to connect factors with zeroes.

Name or initials: ____________   Date: ____________

Expanding (x − a)(x − b) gives x² − (a + b)x + ab.

1. Choose a product. Consider x² − 7x + 12. We need two numbers with sum 7 and product 12.
2. Find the pair. The pair 3 and 4 works because 3 + 4 = 7 and 3 × 4 = 12.
3. Check by expansion. (x − 3)(x − 4) = x² − 4x − 3x + 12 = x² − 7x + 12.
4. Set each factor to zero. x − 3 = 0 or x − 4 = 0, so the zeroes are 3 and 4.

Worked example: For x² + x − 6, the numbers 3 and −2 have sum 1 and product −6. Thus p(x) = (x + 3)(x − 2), so its zeroes are −3 and 2.

1. Which pair contains both zeroes of x² − 5x + 6?

   1. 2 and 3 · 2. -2 and -3 · 3. 5 and 6

2. Which pair contains both zeroes of x² − 5x + 4?

   1. -1 and -4 · 2. 5 and 4 · 3. 1 and 4

3. Which pair contains both zeroes of x² − 8x + 15?

   1. 8 and 15 · 2. 3 and 5 · 3. -3 and -5

Discuss: Create a polynomial with zeroes −2 and 5. Expand (x + 2)(x − 5) and verify both inputs.

Lesson 3 · Relate zeroes to coefficients

Check the sum and product of two zeroes without solving again.

Name or initials: ____________   Date: ____________

A nonzero leading coefficient changes coefficient ratios; it does not allow you to ignore a.

1. Start from factors. If α and β are zeroes, write p(x) = a(x − α)(x − β).
2. Expand the structure. Expansion gives ax² − a(α + β)x + aαβ.
3. Compare coefficients. Matching ax² + bx + c gives b = −a(α + β) and c = aαβ.
4. Divide by a. Since a ≠ 0, α + β = −b/a and αβ = c/a.

Worked example: For 2x² − 10x + 12, a = 2, b = −10 and c = 12. Sum = 10/2 = 5 and product = 12/2 = 6. Zeroes 2 and 3 satisfy both checks.

1. For 2x² + (-10)x + (12), what is the sum of its zeroes?

   1. 5 · 2. 6 · 3. 4

2. For 3x² + (-9)x + (6), what is the product of its zeroes?

   1. 3 · 2. 1 · 3. 2

3. For 2x² + (6)x + (4), what is the sum of its zeroes?

   1. -4 · 2. -3 · 3. -2

Discuss: Write a monic quadratic whose zeroes have sum 9 and product 14. Explain why x² − 9x + 14 works.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. For p(x) = (x − 5)(x − 7), which listed value is a zero?

   15 / 25 / 5

2. For p(x) = (x − 6)(x − 8), which listed value is a zero?

   26 / 6 / 16

3. Which pair contains both zeroes of x² − 9x + 20?

   -4 and -5 / 9 and 20 / 4 and 5

4. Which pair contains both zeroes of x² − 10x + 21?

   10 and 21 / 3 and 7 / -3 and -7

5. For 3x² + (6)x + (-9), what is the sum of its zeroes?

   -1 / -3 / -2

6. For 5x² + (-15)x + (10), what is the product of its zeroes?

   1 / 2 / 3

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. For p(x) = (x − 7)(x − 9), which listed value is a zero?

   7 / 17 / 27

2. For p(x) = (x − 8)(x − 10), which listed value is a zero?

   18 / 28 / 8

3. Which pair contains both zeroes of x² − 11x + 30?

   5 and 6 / -5 and -6 / 11 and 30

4. Which pair contains both zeroes of x² − 12x + 32?

   -4 and -8 / 12 and 32 / 4 and 8

5. For 2x² + (-14)x + (20), what is the sum of its zeroes?

   7 / 8 / 6

6. For 3x² + (12)x + (9), what is the product of its zeroes?

   4 / 2 / 3