Chapter 2: Introduction to Linear Polynomials
Chapter 2: Introduction to Linear Polynomials
Exercise
Question
Question: 11
19 / 26
19 / 26
Let p(x) = ax + b and q(x) = cx + d be two linear polynomials such that:
(i) p(0) = 5
(ii) The polynomial p(x) - q(x) cuts the x-axis at (3, 0).
(iii) The sum p(x) + q(x) is equal to 6x + 4 for all real x.
Find the polynomials p(x) and q(x).
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Answer
1
Substitute x = 0 into p(x) = ax + b and use p(0) = 5 to find b.
\text{${color:#2E2EFF}5{/color} = a × {color:#2E2EFF}0{/color} + b$ $\quad$ ${color:#625B5F}[Given: p(0) = 5]{/color}$}
⇒
\text{ $b = 5$ $\quad$ ${color:#B45309}[Result 1]{/color}$}
2
Expand p(x) + q(x).
$p(x) + q(x) = (ax + b) + (cx + d)$
⇒
$p(x) + q(x) = (a + c)x + (b + d)$
3
Compare p(x) + q(x) with 6x + 4 to find a + c and b + d.
$6x + 4 = (a + c)x + (b + d)$
⇒
\text{ $6x + 4 = (a + c)x + ({color:#2E2EFF}5{/color} + d)$}
Comparing the coefficients of $x$ and the constant terms:
\text{$a + c = 6$ $\quad$ ${color:#6D28D9}[Equation 1]{/color}$}
\text{$5 + d = 4$ $\quad$ ${color:#6D28D9}[Equation 2]{/color}$}
4
Solve Equation 2 to find d.
$5 + d = 4$
⇒
\text{ $d = -1$ $\quad$ ${color:#B45309}[Result 2]{/color}$}
5
Expand p(x) - q(x).
$p(x) - q(x) = (ax + b) - (cx + d)$
⇒
$p(x) - q(x) = (a - c)x + (b - d)$
6
Use the fact that p(x) - q(x) cuts the x-axis at (3, 0) to form Equation 3.
\text{${color:#2E2EFF}0{/color} = (a - c) × {color:#2E2EFF}3{/color} + (b - d)$ $\quad$ ${color:#625B5F}[At the x-intercept, x = 3 and p(x) - q(x) = 0]{/color}$}
⇒
\text{ $0 = (a - c)3 + ({color:#2E2EFF}5{/color} - ({color:#2E2EFF}-1{/color}))$ $\quad$ ${color:#625B5F}[From Result 1, b = 5; from Result 2, d = -1]{/color}$}
⇒
$0 = 3(a - c) + 6$
⇒
\text{ $0 = (a - c) + 2$ $\quad$ ${color:#625B5F}[Divide both sides by 3]{/color}$}
⇒
\text{ $a - c = -2$ $\quad$ ${color:#6D28D9}[Equation 3]{/color}$}
7
Add Equations 1 and 3 to find a.
$(a + c) + (a - c) = 6 + (-2)$
⇒
$2a = 4$
⇒
\text{ $a = 2$ $\quad$ ${color:#B45309}[Result 3]{/color}$}
8
Substitute the value of a into Equation 1 to find c.
$a + c = 6$
⇒
\text{ ${color:#2E2EFF}2{/color} + c = 6$}
⇒
$c = 6 - 2$
⇒
\text{ $c = 4$ $\quad$ ${color:#B45309}[Result 4]{/color}$}
9
Write the polynomials using the values of a, b, c, and d.
Answer: $p(x) = 2x + 5$ and $q(x) = 4x - 1$