Introduction to Linear Polynomials: questions with answers

A constant term has no variable. Both terms contain $x$, so the constant term is 0. The number $-3$ multiplies $x$, so it is the coefficient of $x$.

The term in $x$ is $-7x = (-7)\times x$. Therefore, the coefficient of $x$ is $-7$, including the minus sign. The number $5$ is the constant term, not the coefficient of $x$.

The degree is the highest power of the variable that appears. The powers present here are 7, 2 and 0. The highest of these is 7, so the degree is 7. Adding the powers instead gives 9, which is not the degree.

A bare number is a polynomial of degree 0. The chapter shows why: 3 can be written as $3x^0$, in which the power of the variable is 0, so 0 is the highest power present. Saying it has no degree at all leaves the bottom rung off the ladder whose top — linear, quadratic, cubic — the rest of the chapter is built on.

The amount decreases by Rs 10 each week. Equal changes over equal time intervals make the pattern linear. Since the amounts decrease, the pattern is linear decay, even though the amounts remain positive.

In 4, 8, 16, 32, the successive differences are 4, 8 and 16. These differences are unequal, so this sequence is not a linear pattern. In 5, 10, 15, 20 and 4, 9, 14, 19, each successive difference is 5. In 50, 44, 38, 32, each successive difference is −6. These three sequences are linear patterns because each has a constant difference.

A polynomial is linear when its degree is 1, that is, when the highest power of the variable is 1. $7p$ has degree 1, so it is linear, even though it has only one term. $4p^2 + 5$ has degree 2, so it is not linear, even though it has two terms. Counting terms does not decide the degree. So the number of linear polynomials in the list is 2.

The highest power of $p$ in $6p^2 + 7$ is 2, so it is a quadratic polynomial. Having two terms does not make a polynomial linear. A linear polynomial has degree 1.

It is: $6 \times (-5) = -30$ and $-30 + 4 = -26$. The positive input 5 gives 34 instead, so the sign is not a detail that can be dropped on the way in. The input -36 is what would be needed if the coefficient were ADDED to the input rather than multiplied by it, and it is nowhere near -5.

This is the chess club written as a polynomial: Rs 20 to join, Rs 5 a match, Rs 55 paid in all. Taking the constant off leaves 35, and each match accounts for 5 of it, so $35 \div 5 = 7$. Dividing the whole 55 by 5 gives 11, which counts the joining fee as matches. Taking 5 away instead of dividing by it gives 30 -- subtracting undoes an addition, and the coefficient was never added to $n$, it was multiplied by it.

Losing value is decay, so the coefficient of $n$ is negative: $-900n + 7000$. The purchase price Rs 7000 is the value at $n = 0$, so it stays as the constant. Dropping it leaves $-900n$, which is negative from the first year and cannot be a price. With a plus sign the phone would gain Rs 2700 in three years instead of losing value.

The starting balance was Rs $515 + 5 \times 7 = 550$. After $n$ days, subtract Rs $7n$ from this amount. The rule is $-7n + 550$. A positive coefficient would make the balance increase, while $-7n$ leaves out the starting balance.

Only the constant has changed, from 15 to 60, so the line meets the $y$-axis 45 higher up and everything else about it is the same. The slope is still $-8$: the line still falls by 8 for every step of 1 in $x$, so it is neither steeper nor gentler, and it still shows decay. Sitting higher is not the same as growing -- both lines head downwards, one of them starting further up.

The amount of water falls by 21 litres each hour. The change each hour is $-21$, so the slope is $-21$. That is a negative number, so the sign of the slope is negative. The readings 340, 319, 298 and so on are all positive. But that is the amount in the tank, not the way it changes. The slope is the CHANGE, and the change is downwards.

Distinct parallel lines have the same slope and different y-intercepts. The given line has slope 3 and y-intercept 9. $y = 3x + 10$ has slope 3 and y-intercept 10, so it is parallel to the given line. $y = 7x + 9$ has a different slope, 7, so it is not parallel. $y = 3x + 9$ represents the given line itself.