Question 1(i)
How many zeroes does the polynomial represented by graph (i) have?
Steps only
The graph does not meet the x-axis.
Answer
0 zeroes.
CLASS 10 · MATHEMATICS · CHAPTER 2
Polynomials
Read zeroes from graphs, find the zeroes of quadratic polynomials and build a polynomial from its zeroes. Every subpart has a short worked answer and an optional explanation.
18 worked answersAll 2 exercise sets
What you will learn
- Connect a zero of a polynomial with an x-axis intersection of its graph.
- Verify the sum and product relations for the zeroes of a quadratic polynomial.
- Form a quadratic polynomial when its zeroes are known.
18 worked answers · short steps first, explanations when you need them
Exercise 2.1
Textbook pages 8
Question 1(ii)
How many zeroes does the polynomial represented by graph (ii) have?
Steps only
The graph meets the x-axis once.
Answer
1 zero.
Question 1(iii)
How many zeroes does the polynomial represented by graph (iii) have?
Steps only
The graph meets the x-axis three times.
Answer
3 zeroes.
Question 1(iv)
How many zeroes does the polynomial represented by graph (iv) have?
Steps only
The graph meets the x-axis twice.
Answer
2 zeroes.
Question 1(v)
How many zeroes does the polynomial represented by graph (v) have?
Steps only
The graph meets the x-axis four times.
Answer
4 zeroes.
Question 1(vi)
How many zeroes does the polynomial represented by graph (vi) have?
Steps only
The graph meets the x-axis three times.
Answer
3 zeroes.
Exercise 2.2
Textbook pages 12–13
Question 1(i)
Find the zeroes of x² − 2x − 8 and verify the relation between the zeroes and coefficients.
Steps only
x² − 2x − 8 = (x − 4)(x + 2)
Zeroes: 4, −2
4 + (−2) = 2 = −b/a; 4(−2) = −8 = c/a
Answer
The zeroes are 4 and −2; both relations are verified.
Question 1(ii)
Find the zeroes of 4s² − 4s + 1 and verify the relation between the zeroes and coefficients.
Steps only
4s² − 4s + 1 = (2s − 1)²
Zeroes: 1/2, 1/2
1/2 + 1/2 = 1 = −b/a; (1/2)(1/2) = 1/4 = c/a
Answer
Both zeroes are 1/2; both relations are verified.
Question 1(iii)
Find the zeroes of 6x² − 7x − 3 and verify the relation between the zeroes and coefficients.
Steps only
6x² − 7x − 3 = (3x + 1)(2x − 3)
Zeroes: −1/3, 3/2
−1/3 + 3/2 = 7/6 = −b/a; (−1/3)(3/2) = −1/2 = c/a
Answer
The zeroes are −1/3 and 3/2; both relations are verified.
Question 1(iv)
Find the zeroes of 4u² + 8u and verify the relation between the zeroes and coefficients.
Steps only
4u² + 8u = 4u(u + 2)
Zeroes: 0, −2
0 + (−2) = −2 = −b/a; 0(−2) = 0 = c/a
Answer
The zeroes are 0 and −2; both relations are verified.
Question 1(v)
Find the zeroes of t² − 15 and verify the relation between the zeroes and coefficients.
Steps only
t² − 15 = (t − √15)(t + √15)
Zeroes: √15, −√15
Sum = 0 = −b/a; product = −15 = c/a
Answer
The zeroes are √15 and −√15; both relations are verified.
Question 1(vi)
Find the zeroes of 3x² − x − 4 and verify the relation between the zeroes and coefficients.
Steps only
3x² − x − 4 = (3x − 4)(x + 1)
Zeroes: 4/3, −1
4/3 − 1 = 1/3 = −b/a; (4/3)(−1) = −4/3 = c/a
Answer
The zeroes are 4/3 and −1; both relations are verified.
Question 2(i)
Find a quadratic polynomial whose zeroes have sum 1/4 and product −1.
Steps only
p(x) = x² − (1/4)x − 1
Multiply by 4: p(x) = 4x² − x − 4
Answer
One such polynomial is 4x² − x − 4.
Question 2(ii)
Find a quadratic polynomial whose zeroes have sum √2 and product 1/3.
Steps only
p(x) = x² − √2x + 1/3
Multiply by 3: p(x) = 3x² − 3√2x + 1
Answer
One such polynomial is 3x² − 3√2x + 1.
Question 2(iii)
Find a quadratic polynomial whose zeroes have sum 0 and product √5.
Steps only
p(x) = x² − 0x + √5
Answer
One such polynomial is x² + √5.
Question 2(iv)
Find a quadratic polynomial whose zeroes have sum 1 and product 1.
Steps only
p(x) = x² − x + 1
Answer
One such polynomial is x² − x + 1.
Question 2(v)
Find a quadratic polynomial whose zeroes have sum −1/4 and product 1/4.
Steps only
p(x) = x² + (1/4)x + 1/4
Multiply by 4: p(x) = 4x² + x + 1
Answer
One such polynomial is 4x² + x + 1.
Question 2(vi)
Find a quadratic polynomial whose zeroes have sum 4 and product 1.
Steps only
p(x) = x² − 4x + 1
Answer
One such polynomial is x² − 4x + 1.