# MR02. Math Refresher — Polynomials

## Polynomials

A polynomial is a collection of terms joined by \(+\) and \(–\) operators. Polynomials with one term are called

*monomials,*with two terms is called

*binomial,*with three terms is a

*trinomial,*and so on.

Another (and more important) way to classify a polynomial is by its *degree.* The degree is the value of the highest exponent (power) in the polynomial. For example:

$$ x^2 + 2x – 2$$

is a second-degree polynomial. Similarly,

$$y^7 – y^5 + 2$$

is a seventh-degree polynomial.

Finally, each term can be “named” by its exponent. If a term has the exponent 2, it is called the “square” term, exponent 3 is the “cubic” term, exponent 4, the “quartic” term, and so on.

Next we will cover multiplying two binomials.

Got all that? Let’s go…

## Multiplying

Multiplying binomials is taught with an acronymn — FOIL — first, outer, inner, last. How does this work?

$$ (a + b)(c + d)$$

- The first terms: \(( {\bf a} + b) ({\bf c} + d)\) are \(a\) and \(c,\) so their product is \(F = ac.\)

- The outer terms: \(({\bf a} + b) (c + {\bf d})\) are \(a\) and \(d,\) so \(O = ad.\)

- The inner terms: \((a + {\bf b}) ({\bf c} + d)\) are \(b\) and \(c,\) so \(I = bc.\)

- The last terms: \((a + {\bf b}) (c + {\bf d})\) are \(b\) and \(d,\) so \(L = bd.\)

Now, sum them all together into the expanded polynomial:

$$ac + ad + bc + bd .$$

Let’s try some examples.

**Examples.**

- Using the FOIL technique, multiply out \((x + 3)(x + 4)\)

$$ = x^2 + 4x + 3x + 12$$

$$ = x^2 + 7x + 12 .$$

- Use FOIL to multiply \((x + 6)(x – 5)\)

$$ = x^2 – 5x + 6x – 30$$

$$ = x^2 + x – 30 .$$

### Squaring

This is a place where students get tripped up. Note that

$$ (a + b)^2 \neq a^2 + b^2 {\rm !}$$

Here’s the correct way:

$$ (a + b)^2 = (a + b)(a + b)$$

Now, FOIL it out, giving:

$$ a^2 + ab + ab + b^2$$

which equals

$$a^2 + 2ab + b^2 .$$

**Examples.**

- Square \((x + 4)\)

$$(x + 4)(x + 4) = x^2 + 4x + 4x + 16 = x^2 + 8x + 16$$

- Square \((x – 4)\)

$$(x – 4)(x – 4) = x^2 – 4x – 4x + 16 = x^2 – 8x + 16$$

### Special cases

You just saw the first special case:

$$ (a + b)^2 = a^2 + 2ab + b^2$$

Next, learn this one:

$$(a + b)(a – b) = a^2 – ab + ab – b^2$$

The \(ab\) terms cancel each other, leaving:

$$(a + b)(a – b) = a^2 – b^2$$

So, remember this: the difference of two squares \((a^2 – b^2)\) can be factored into \((a + b)(a – b).\)

**Examples.**

- Multiply \((x + 3)(x – 3)\)

$$(x + 3)(x – 3) = x^2 – 3x + 3x – 9 = x^2 – 9$$ - Multiply \((x + 5)(x – 5)\)

$$(x + 5)(x – 5) = x^2 – 5x + 5x – 25 = x^2 – 25$$

## Squares and Square Roots

Here’s a good place to mention this. Question: What is \(\sqrt 4\) ? Immediate answer: 2 !

But that’s not correct. What? The correct answer: \(\sqrt 4 = \pm 2.\) Why? Because \((-2)^2 = 4,\) also.

When you take a square root, always think “\(\pm\)”.

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Thanks for these. This is where we are with our Math….

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— Doc