Introduction to Linear Polynomials | EOT

Question 2

Find the values of the following polynomials at the indicated values of the variables.

(i) $5x^2 - 3x + 7$ if $x = 1$

(ii) $4t^3 - t^2 + 6$ if $t = a$

Solution

Understand the Question

To find the value of a polynomial at a given value of a variable, substitute the given value in place of the variable everywhere it appears.
Then, evaluate powers first, perform multiplications, and simplify using standard order of operations.

(i) Find the value of $5x^2 - 3x + 7$ at $x = 1$

Step 1 · Substitute $x = 1$ into the Polynomial

Let $p(x) = 5x^2 - 3x + 7$ .

Substitute $x = 1$ :

\begin{aligned} p(1) &= 5(1)^2 - 3(1) + 7 \\ &= 5(1) - 3 + 7 \\ &= 5 - 3 + 7 \\ &= 9 \end{aligned}

Answer

(i) $9$

(ii) Find the value of $4t^3 - t^2 + 6$ at $t = a$

Step 1 · Substitute $t = a$ into the Polynomial

Let $q(t) = 4t^3 - t^2 + 6$ .

Substitute $t = a$ :

\begin{aligned} q(a) &= 4(a)^3 - (a)^2 + 6 \\ &= 4a^3 - a^2 + 6 \end{aligned}

Answer

(ii) $4a^3 - a^2 + 6$

Common Mistakes

Order of Operations: Evaluating $5(1)^2$ by multiplying first instead of exponentiating first. Always compute the power before multiplying with the coefficient.
Variable Substitution: Confusing variable replacement in part (ii) as something that needs to be solved for a numerical value. Replacing variable $t$ with constant $a$ simply yields the polynomial expression in terms of $a$ .

More questions in EOT

Write a polynomial of degree 3 in the variable $x$ , in which the coefficient of the $x^2$ term is $-7$ .

Find the values of the following polynomials at the indicated values of the variables.

(i) $5x^2 - 3x + 7$ if $x = 1$

(ii) $4t^3 - t^2 + 6$ if $t = a$

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Q10

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

(i) Find the polynomial $p(x)$ .

(ii) Find the coordinates where the graph of $p(x)$ cuts the axes.

(iii) Draw the graph of $p(x)$ and verify your answers.

Q11

*11. 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 $q(x)$ .

Q12

*12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage.

(i) Draw the next two stages of the pattern. How many matchsticks will be required at these stages?

(ii) Complete the following table.

(iii) Find a rule to determine the number of matchsticks required for the $n^{\text{th}}$ stage.

(iv) How many matchsticks will be required for the $15^{\text{th}}$ stage of the pattern?

(v) Can 200 matchsticks form a stage in this pattern? Justify your answer.

Q13

*13. Let $p(x) = ax + b$ and $q(x) = cx + d$ be two linear polynomials such that:

(i) The graph of $p(x)$ passes through the points $(2, 3)$ and $(6, 11)$ .

(ii) The graph of $q(x)$ passes through the point $(4, -1)$ .

(iii) The graph of $q(x)$ is parallel to the graph of $p(x)$ .

Find the polynomials $p(x)$ and $q(x)$ . Also, find the coordinates of the point where these lines meet the $x$ -axis.

Q14

*14. What do all linear functions of the form $f(x) = ax + a, a > 0$ , have in common?