GRADE 10 · A4 PRINT VIEW
Polynomials · Answer guide
Use your browser Print command and select A4. Questions and answers have separate pages.
Lesson 1 · Zeroes and the x-axis
Connect a zero, an input value and an x-intercept.
For p(x) = (x − 1)(x − 3), which listed value is a zero?
Answer: 1. At x = 1, the first factor is zero, so p(1) = 0. The other zero is 3.
For p(x) = (x − 2)(x − 4), which listed value is a zero?
Answer: 2. At x = 2, the first factor is zero, so p(2) = 0. The other zero is 4.
For p(x) = (x − 3)(x − 5), which listed value is a zero?
Answer: 3. At x = 3, the first factor is zero, so p(3) = 0. The other zero is 5.
Discuss: Choose (x − 2)(x − 5). Make a table for x = 0 through 6 and mark the two zero outputs.
Teaching note: A zero is an x-value, not the y-intercept. A graph can touch the x-axis without crossing it. Open responses need discussion; they are not automatically marked.
Lesson 2 · Factor a quadratic to find its zeroes
Use multiplication to connect factors with zeroes.
Which pair contains both zeroes of x² − 5x + 6?
Answer: 2 and 3. x² − 5x + 6 = (x − 2)(x − 3). A zero product has at least one zero factor.
Which pair contains both zeroes of x² − 5x + 4?
Answer: 1 and 4. x² − 5x + 4 = (x − 1)(x − 4). A zero product has at least one zero factor.
Which pair contains both zeroes of x² − 8x + 15?
Answer: 3 and 5. x² − 8x + 15 = (x − 3)(x − 5). A zero product has at least one zero factor.
Discuss: Create a polynomial with zeroes −2 and 5. Expand (x + 2)(x − 5) and verify both inputs.
Teaching note: Factoring is not dividing the whole expression by x. Always expand your proposed factors to check the signs. Open responses need discussion; they are not automatically marked.
Lesson 3 · Relate zeroes to coefficients
Check the sum and product of two zeroes without solving again.
For 2x² + (-10)x + (12), what is the sum of its zeroes?
Answer: 5. For ax² + bx + c, the sum is −b/a and product is c/a. Here the requested value is 5.
For 3x² + (-9)x + (6), what is the product of its zeroes?
Answer: 2. For ax² + bx + c, the sum is −b/a and product is c/a. Here the requested value is 2.
For 2x² + (6)x + (4), what is the sum of its zeroes?
Answer: -3. For ax² + bx + c, the sum is −b/a and product is c/a. Here the requested value is -3.
Discuss: Write a monic quadratic whose zeroes have sum 9 and product 14. Explain why x² − 9x + 14 works.
Teaching note: The sum has a minus sign; the product does not. Both relations divide by a. Open responses need discussion; they are not automatically marked.
Final check · Set 1
Try one set after the lessons. Use the other set after reviewing any errors.
For p(x) = (x − 5)(x − 7), which listed value is a zero?
5. At x = 5, the first factor is zero, so p(5) = 0. The other zero is 7.
For p(x) = (x − 6)(x − 8), which listed value is a zero?
6. At x = 6, the first factor is zero, so p(6) = 0. The other zero is 8.
Which pair contains both zeroes of x² − 9x + 20?
4 and 5. x² − 9x + 20 = (x − 4)(x − 5). A zero product has at least one zero factor.
Which pair contains both zeroes of x² − 10x + 21?
3 and 7. x² − 10x + 21 = (x − 3)(x − 7). A zero product has at least one zero factor.
For 3x² + (6)x + (-9), what is the sum of its zeroes?
-2. For ax² + bx + c, the sum is −b/a and product is c/a. Here the requested value is -2.
For 5x² + (-15)x + (10), what is the product of its zeroes?
2. For ax² + bx + c, the sum is −b/a and product is c/a. Here the requested value is 2.
Final check · Set 2
Try one set after the lessons. Use the other set after reviewing any errors.
For p(x) = (x − 7)(x − 9), which listed value is a zero?
7. At x = 7, the first factor is zero, so p(7) = 0. The other zero is 9.
For p(x) = (x − 8)(x − 10), which listed value is a zero?
8. At x = 8, the first factor is zero, so p(8) = 0. The other zero is 10.
Which pair contains both zeroes of x² − 11x + 30?
5 and 6. x² − 11x + 30 = (x − 5)(x − 6). A zero product has at least one zero factor.
Which pair contains both zeroes of x² − 12x + 32?
4 and 8. x² − 12x + 32 = (x − 4)(x − 8). A zero product has at least one zero factor.
For 2x² + (-14)x + (20), what is the sum of its zeroes?
7. For ax² + bx + c, the sum is −b/a and product is c/a. Here the requested value is 7.
For 3x² + (12)x + (9), what is the product of its zeroes?
3. For ax² + bx + c, the sum is −b/a and product is c/a. Here the requested value is 3.
Original learning companion, not full textbook exercise solutions. Three lessons cover zeroes, simple quadratic factorisation and coefficient relations. A full range of graph shapes and every textbook exercise remain outside this companion.