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Polynomials

Algebra · level 6, lesson 2 of 6

A question to start

Work out \(23 \times 47\) by splitting each number into tens and ones:

\[23 \times 47 = (20 + 3)(40 + 7).\]

Draw a rectangle 23 by 47, and cut it into four pieces: \(20 \times 40 = 800\), \(20 \times 7 = 140\), \(3 \times 40 = 120\) and \(3 \times 7 = 21\). Add them: \(800 + 140 + 120 + 21 = 1081\).

Now look at \((2x + 3)(4x + 7)\). Cut its rectangle the same way: \(8x^2 + 14x + 12x + 21 = 8x^2 + 26x + 21\). Put \(x = 10\) in and you get \(800 + 260 + 21 = 1081\) again. Of course: when \(x = 10\), \(2x + 3\) is 23 and \(4x + 7\) is 47.

The long multiplication you learned in primary school is algebra in disguise, with \(x = 10\). In this lesson you meet the expressions that this multiplication produces, polynomials, and learn to add, subtract and multiply them.

The idea

The words

A monomial is a single term made of a number times letters with whole-number exponents: \(7\), \(-3x\), \(5x^2y\), \(\tfrac12 a^3\). The number in front is the coefficient.

A polynomial is a sum of monomials, such as \(4x^3 - x^2 + 7x - 2\). Its pieces are its terms. Special names count the terms:

Number of terms Name Example
1 monomial \(6x^2\)
2 binomial ("two names") \(x + 5\)
3 trinomial \(x^2 - 5x + 6\)
more polynomial \(x^4 + x^3 - 2x + 1\)

A term with no letter, like the \(-2\), is the constant term.

Degree

The degree of a term is the total of its exponents: \(5x^2y\) has degree \(2 + 1 = 3\), and a constant like 7 has degree 0 (think of it as \(7x^0\)). The degree of a polynomial is the highest degree of any of its terms. Degree 1 is called linear (\(3x + 1\)), degree 2 quadratic (\(x^2 - 5x + 6\)), degree 3 cubic, degree 4 quartic.

A polynomial in one letter is in standard form when its terms are written from the highest power down: \(-x^2 + 4x^3 + 7 - 2x\) becomes \(4x^3 - x^2 - 2x + 7\). Standard form makes degree and coefficients easy to read, and makes like terms easy to line up.

What is not a polynomial

Exponents in a polynomial must be whole numbers 0, 1, 2, .... So these are not polynomials: \(\tfrac{1}{x}\) (which is \(x^{-1}\)), \(\sqrt{x}\) (which is \(x^{1/2}\), as you will see in level 7), and \(2^x\) (the letter is in the exponent).

Adding and subtracting

Adding polynomials is just collecting like terms (level 4): \(x^2\) terms with \(x^2\) terms, \(x\) terms with \(x\) terms, numbers with numbers.

\[(3x^2 - 2x + 5) + (x^2 + 7x - 8) = 4x^2 + 5x - 3.\]

Subtracting a polynomial means subtracting every term of it. The minus in front of the bracket flips every sign inside (it is \(-1\) times the bracket, by the distributive law):

\[(5a^2 + 3a - 1) - (2a^2 - 4a + 6) = 5a^2 + 3a - 1 - 2a^2 + 4a - 6 = 3a^2 + 7a - 7.\]

Multiplying a monomial by a polynomial

Use the distributive law, and the exponent laws for each product:

\[-2x(3x^2 - x + 4) = -6x^3 + 2x^2 - 8x.\]

Each term of the bracket is multiplied by \(-2x\): \(-2x \times 3x^2 = -6x^3\), \(-2x \times (-x) = +2x^2\), \(-2x \times 4 = -8x\).

Multiplying two binomials: the area model

To multiply \((x + 3)(x + 5)\), draw a rectangle whose sides are \(x + 3\) and \(x + 5\). Split each side into its two parts. The rectangle splits into four smaller rectangles, and the total area is the sum of the four.

x3 x5 x²3x 5x15 (x + 3)(x + 5) = x² + 3x + 5x + 15 = x² + 8x + 15
\[(x + 3)(x + 5) = x^2 + 3x + 5x + 15 = x^2 + 8x + 15.\]

The two middle pieces are like terms, so they combine. That is why the answer has three terms, not four.

Why every term meets every term

The area model is the distributive law used twice. Treat \((x + 5)\) as a single number first:

\[(x + 3)(x + 5) = x(x + 5) + 3(x + 5) = x^2 + 5x + 3x + 15.\]

Each term of the first bracket multiplies each term of the second. That is the rule for multiplying any two polynomials: every term of one meets every term of the other once. Two terms times two terms gives \(2 \times 2 = 4\) products; two terms times three terms gives 6; and so on. A grid with one row per term of the first bracket and one column per term of the second keeps track of them, even when there are negative terms (the "areas" are then just products, some negative).

FOIL: a shortcut, not a rule

For two binomials, people often remember the four products as FOIL: First terms, Outer terms, Inner terms, Last terms. For \((2x - 3)(x + 4)\): First \(2x \cdot x = 2x^2\), Outer \(2x \cdot 4 = 8x\), Inner \(-3 \cdot x = -3x\), Last \(-3 \cdot 4 = -12\). Total \(2x^2 + 5x - 12\).

FOIL is just a way of not forgetting one of the four rectangles. It only works for binomial times binomial. For \((x + 2)(x^2 - 3x + 1)\) there are six products, and FOIL has no letters for them, so use the grid or the "every term meets every term" rule.

Three special products

Three products come up so often that they are worth knowing on sight:

\[(a + b)^2 = a^2 + 2ab + b^2, \qquad (a - b)^2 = a^2 - 2ab + b^2, \qquad (a + b)(a - b) = a^2 - b^2.\]

The picture for the first: a square of side \(a + b\) splits into an \(a\) by \(a\) square, a \(b\) by \(b\) square, and two \(a\) by \(b\) rectangles.

abab a²ababb² (a + b)² = a² + 2ab + b²

The third one is surprising: the middle terms cancel. \((a + b)(a - b) = a^2 - ab + ab - b^2 = a^2 - b^2\). It gives a mental arithmetic trick: \(53 \times 47 = (50 + 3)(50 - 3) = 2500 - 9 = 2491\).

Degrees add when you multiply

When you multiply two polynomials, the highest-degree term of the answer comes from multiplying the two highest-degree terms, and exponents add. So a degree 2 polynomial times a degree 3 polynomial has degree 5. This is a quick check: \((x + 2)(x^2 - 3x + 1)\) must have degree \(1 + 2 = 3\).

Worked examples

1. Adding. \((3x^2 - 2x + 5) + (x^2 + 7x - 8) = 4x^2 + 5x - 3\).

  • Check with \(x = 1\): \(6 + 0 = 6\), and \(4 + 5 - 3 = 6\) ✓.

2. Subtracting. \((5a^2 + 3a - 1) - (2a^2 - 4a + 6) = 3a^2 + 7a - 7\) (shown above).

  • Check with \(a = 1\): \(7 - 4 = 3\), and \(3 + 7 - 7 = 3\) ✓.

3. Two binomials with negatives. Expand \((2x - 3)(x + 4)\).

\(x\) \(+4\)
\(2x\) \(2x^2\) \(8x\)
\(-3\) \(-3x\) \(-12\)
  • Add the four cells: \(2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12\).
  • Check with \(x = 2\): \((1)(6) = 6\), and \(8 + 10 - 12 = 6\) ✓.

4. Binomial times trinomial. Expand \((x + 2)(x^2 - 3x + 1)\).

  • \(x\) times each: \(x^3 - 3x^2 + x\). Then \(2\) times each: \(2x^2 - 6x + 2\).
  • Collect: \(x^3 - x^2 - 5x + 2\). (Degree 3, as predicted.)
  • Check with \(x = 1\): \(3 \times (-1) = -3\), and \(1 - 1 - 5 + 2 = -3\) ✓.

5. Special products. \((x - 4)^2 = x^2 - 8x + 16\) and \((x + 6)(x - 6) = x^2 - 36\).

  • In the first, the middle term is \(2 \times x \times (-4) = -8x\), and the last is \((-4)^2 = +16\).

6. Expand and simplify. \((x + 1)(x + 2) - (x - 3)^2\).

  • \((x + 1)(x + 2) = x^2 + 3x + 2\) and \((x - 3)^2 = x^2 - 6x + 9\).
  • Subtract (flip every sign of the second): \(x^2 + 3x + 2 - x^2 + 6x - 9 = 9x - 7\). The \(x^2\) terms cancel.
  • Check with \(x = 0\): \(2 - 9 = -7\) ✓.

Try it

  1. Give the degree of each and say whether it is a monomial, binomial or trinomial: (a) \(4x^3 - x + 2\); (b) \(7a^2b^3\); (c) \(5\).
  2. Simplify \((2x^2 + 3x - 4) + (5x^2 - x + 9)\).
  3. Simplify \((4y^2 - 3y + 2) - (y^2 + 5y - 7)\).
  4. Expand \(3a(2a^2 - 5a + 1)\).
  5. Expand: (a) \((x + 7)(x - 2)\); (b) \((3x - 1)(2x + 5)\).
  6. Expand \((2x + 3)^2\), and draw the area picture.
  7. Expand \((x - 1)(x^2 + x + 1)\). What do you notice?
  8. Use \((a + b)(a - b) = a^2 - b^2\) to work out \(102 \times 98\) in your head.
Answers
  1. (a) Degree 3, trinomial. (b) Degree \(2 + 3 = 5\), monomial. (c) Degree 0, monomial (a constant).
  2. \(7x^2 + 2x + 5\).
  3. Flip the signs of the second bracket: \(4y^2 - 3y + 2 - y^2 - 5y + 7 = 3y^2 - 8y + 9\). Check with \(y = 1\): \(3 - (-1) = 4\), and \(3 - 8 + 9 = 4\) ✓.
  4. \(6a^3 - 15a^2 + 3a\).
  5. (a) \(x^2 - 2x + 7x - 14 = x^2 + 5x - 14\). (b) \(6x^2 + 15x - 2x - 5 = 6x^2 + 13x - 5\). Check (b) with \(x = 1\): \(2 \times 7 = 14\), and \(6 + 13 - 5 = 14\) ✓.
  6. \((2x + 3)^2 = 4x^2 + 12x + 9\): a \(2x\) by \(2x\) square (\(4x^2\)), two \(2x\) by 3 rectangles (\(6x\) each) and a 3 by 3 square (9). Not \(4x^2 + 9\).
  7. \(x^3 + x^2 + x - x^2 - x - 1 = x^3 - 1\). Almost everything cancels in pairs. (This is the "difference of cubes" pattern, which you will factor in later levels.)
  8. \(102 \times 98 = (100 + 2)(100 - 2) = 10000 - 4 = 9996\).

Common mistake

Forgetting the middle term. \((x + 5)^2\) is not \(x^2 + 25\). Write it as \((x + 5)(x + 5)\) and you get four products: \(x^2 + 5x + 5x + 25 = x^2 + 10x + 25\). The picture shows the two rectangles that the short version leaves out. Test with \(x = 1\): \(36\) against \(26\).

Only flipping the first sign when subtracting. \((4y^2 - 3y + 2) - (y^2 + 5y - 7)\) needs all three signs of the second bracket changed: \(-y^2 - 5y + 7\). A useful habit is to rewrite the subtraction as adding the flipped polynomial before collecting.

Where it leads