Chapter 2: Introduction to Linear Polynomials

Exercise

Question

Question: 10 (i)

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The graph of a linear polynomial p(x) passes through the points (1, 5) and (3, 11).

(i) Find the polynomial p(x).

Answer

1
Let the linear polynomial be p(x) = ax + b, where a and b are constants. Use the point (1, 5) to form Equation 1.
\text{${color:#2E2EFF}5{/color} = a × {color:#2E2EFF}1{/color} + b$}
⇒
\text{ $5 = a + b$ $\quad$ ${color:#6D28D9}[Equation 1]{/color}$}
2
Use the point (3, 11) to form Equation 2.
\text{${color:#2E2EFF}11{/color} = a × {color:#2E2EFF}3{/color} + b$}
⇒
\text{ $11 = 3a + b$ $\quad$ ${color:#6D28D9}[Equation 2]{/color}$}
3
Subtract Equation 1 from Equation 2 to find a.
$11 - 5 = (3a + b) - (a + b)$
⇒
$6 = (3a - a) + (b - b)$
⇒
$6 = 2a$
⇒
\text{ $a = 3$ $\quad$ ${color:#625B5F}[Divide both sides by 2]{/color}$}
4
Substitute the value of a into Equation 1 to find b.
\text{$5 = {color:#2E2EFF}3{/color} + b$}
⇒
$b = 5 - 3$
⇒
$b = 2$
5
Write the polynomial.
Answer: $p(x) = 3x + 2$
Reason: Put $a = 3$ and $b = 2$ in the $p(x) = ax + b$ to get the polynomial.
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