Polynomials · Class 10 Mathematics: quick revision, 51 bites

What makes an expression a polynomial

Whole powers only

In a polynomial the variable appears only with whole-number powers: $x$, $x^2$, $x^3$, …

Takeaway. A variable under a root or in a denominator is the giveaway.

Degree

Look for the highest power

The degree is the highest power of the variable, wherever it sits in the expression.

Takeaway. Scan every term before answering.

Linear, quadratic, cubic

Three degrees, three names

Takeaway. The condition a ≠ 0 is part of each definition.

The value p(k)

Put the number in

$p(x) = x^2 - 3x - 4$.

Takeaway. p(k) is a value, not a zero.

Substituting a negative number

Brackets save marks

$p(-1) = (-1)^2 - 3(-1) - 4 = 1 + 3 - 4 = 0$.

Takeaway. Write the brackets every time.

What a zero is

The number that switches it off

k is a zero of p(x) when $p(k) = 0$.

Takeaway. A zero is an input; the value is the output.

Two different things

p(0) is not 'the zero'

For $p(x) = x^2 - 3x - 4$: $p(0) = -4$, so 0 is NOT a zero.

Takeaway. Zero of p = input giving output 0.

Zero of a linear polynomial

Already inside the coefficients

$2x + 3$ has zero $-\frac{3}{2}$; $5x - 20$ has zero 4.

Takeaway. A linear polynomial has exactly one zero.

How many zeroes

At most, never exactly

Degree n means at most n zeroes: quadratic at most 2, cubic at most 3.

Takeaway. 'At most' is the word examiners test.

Checking a candidate zero

Substitute and look

Takeaway. One substitution answers the question.

Zero or root?

Same number, two names

The zeroes of the polynomial $x^2 - 3x - 4$ are the roots of the equation $x^2 - 3x - 4 = 0$.

Takeaway. Polynomial → zero. Equation → root.

Coefficients can be anything

Surds are allowed in front

$\sqrt{2}x^2 - 3x + \frac{1}{7}$ is a quadratic polynomial.

Takeaway. Only the powers must be whole numbers.

When a term must vanish

A hidden degree question

"If $(k-1)x^2 + 3x + 2$ is linear, find k."

Takeaway. Set the unwanted coefficient to 0.

Constant polynomials

Degree 0 and the odd one out

5 is a polynomial of degree 0 with no zero.

Takeaway. A non-zero constant is never 0.

Zeroes in real life

When the ball lands

A ball's height after t seconds is a quadratic in t. Its zeroes are the times when the height is 0 — the throw and the landing.

Takeaway. A zero is the moment a quantity becomes nothing.

How answers are marked

Show the substitution

For 'verify that 2 is a zero', write $p(2) = \ldots = 0$ in full. The substitution line carries the mark, not the word 'yes'.

Takeaway. Working first, conclusion second.

α, β and γ

Greek names for the answers

The zeroes of a quadratic are usually called α and β; a cubic adds γ.

Takeaway. You will meet them again in topic 3.

A zero is a place on the graph

Where the curve meets the axis

On the x-axis every point has $y = 0$.

Takeaway. Zeroes are x-coordinates of the crossings.

The graph of a linear polynomial

One line, one crossing

Takeaway. A line meets the axis exactly once.

Which way a parabola opens

Smile or frown

The graph of $ax^2 + bx + c$ is a parabola.

Takeaway. Only the sign of a decides the direction.

How a parabola meets the axis

Three ways and no more

Takeaway. Two, one or none — never three.

The y-intercept is not a zero

The other axis does not count

$y = x^2 - 3x - 4$ crosses the y-axis at $(0, -4)$ — that is $p(0)$, the constant term.

Takeaway. Zeroes live on the x-axis only.

Count crossings, not bends

Turns are not zeroes

A parabola has one turning point whether it has two zeroes, one or none.

Takeaway. Only meetings with the x-axis count.

Reading 'how many zeroes?'

Four steps, one mark

Takeaway. That is the whole of Exercise 2.1.

A repeated zero

Touch, don't cut

$y = (x - 2)^2$ touches the axis at $x = 2$ and turns back.

Takeaway. A touch means two equal zeroes.

At most n crossings

The picture proves the limit

A degree-n graph meets the x-axis at most n times, so p(x) has at most n zeroes.

Takeaway. Degree limits crossings; crossings are zeroes.

A cubic that meets the axis three times

Three zeroes on one curve

$y = x^3 - 4x = x(x-2)(x+2)$ meets the axis at −2, 0 and 2.

Takeaway. The most a cubic can manage is three.

Fewer zeroes than the degree

Cubics can do less

Takeaway. Three is a maximum, not a promise.

Three crossings is never a quadratic

A favourite trap

If a graph meets the x-axis at three points, the polynomial has degree at least 3.

Takeaway. Count first, then judge the degree.

Parabolas outside the book

The path of a thrown ball

A ball thrown upwards traces a parabola. Its zeroes are the moments when its height is 0 — the throw and the landing.

Takeaway. 'Parabola' is Greek for 'thrown beside'.

What NCERT asks of you

Reading, not drawing

A footnote says plotting quadratic and cubic graphs is not meant to be done by students and is not assessed.

Takeaway. Learn the shapes; you will not be asked to plot.

x-intercepts vs y-intercept

Same graph, two questions

x-intercepts → the zeroes. y-intercept → the constant term p(0).

Takeaway. Which axis is named decides the answer.

The accessible version

When there is no picture

Figure questions come with a wording-only alternative, such as 'a cubic polynomial has at most how many zeroes?'

Takeaway. The idea is what is tested, not the drawing.

Two identical shapes, different zeroes

Shape is not everything

$y = x^2 + 2$ and $y = x^2 - 4$ are the same shape, but one misses the axis and the other cuts it twice.

Takeaway. Where the curve sits matters as much as its shape.

The two relations

Read them off the coefficients

Takeaway. No need to find the zeroes first.

Where the sign goes

One minus, and only one

The SUM carries the minus: $-\frac{b}{a}$.

Takeaway. Sum is sour — it takes the minus.

Checking with 2x² − 8x + 6

Both sides agree

Takeaway. Divide by a; do not forget it when a ≠ 1.

Splitting the middle term

Product ac, sum b

Takeaway. Factors of ac, never factors of b.

Building a quadratic

Run it backwards

Sum −3 and product 2 give $x^2 + 3x + 2$.

Takeaway. Minus before the sum, plus before the product.

k can be anything

Many answers, all correct

$x^2 - 5x + 6$ and $10x^2 - 50x + 60$ have the same zeroes, 2 and 3.

Takeaway. A multiple never changes the zeroes.

Why the relations hold

Compare the coefficients

$ax^2 + bx + c = k(x - \alpha)(x - \beta) = k[x^2 - (\alpha + \beta)x + \alpha\beta]$.

Takeaway. The proof is one comparison of coefficients.

3x² + 5x − 2

Fractions are fine

$(3x - 1)(x + 2)$, so the zeroes are $\frac13$ and −2.

Takeaway. Verification earns its own marks.

x² − 3

Even with surds

$x^2 - 3 = (x - \sqrt3)(x + \sqrt3)$: the zeroes are $\pm\sqrt3$.

Takeaway. The relations hold for any kind of zero.

Useful rewrites

Without finding α and β

Takeaway. Turn the expression into sum and product, then substitute.

α² + β² in one line

13 without the zeroes

For $x^2 - 5x + 6$: sum 5, product 6.

Takeaway. Rewrite first, substitute second.

Finding a missing coefficient

Set it equal and solve

"The sum of the zeroes of $kx^2 - 3x + 5$ is 1."

Takeaway. Write the relation, then solve for the letter.

Two numbers, one quadratic

A rectangle in disguise

Takeaway. 'Sum and product' questions are quadratics in disguise.

When a is not 1

Divide by a

For $2x^2 - 8x + 6$ the sum is $\frac{8}{2} = 4$, not 8.

Takeaway. Both relations divide by the coefficient of x².

x² − (sum)x + (product)

The shape to remember

Write the frame first, then drop the numbers in: $x^2 - Sx + P$.

Takeaway. The frame prevents sign slips.

Where the relations come from

Vieta, 400 years ago

These are the simplest of Vieta's formulas, from the 16th century. Higher degrees have their own versions, outside Class 10.

Takeaway. An old idea that still saves time.

How the 3-mark answer is marked

Say which is which

Takeaway. Both comparisons carry marks, not just the zeroes.