Math: Expressions and Polynomials

This is Unit Eight of ACT Preparation: expressions and polynomials. Unit seven worked with numbers; this unit works with expressions, where the same rules apply to letters. One habit runs through it: every rewrite must keep the value the same, and a check by substitution can prove a mistake. By the end you will be able to add, subtract and multiply polynomials. To expand products with the special patterns. To factor completely. To simplify and combine rational expressions, keeping their excluded values. And to recognize an equivalent expression.

Six entries on polynomials and their products.

The parts of a polynomial.

ACT 0701 Polynomial Vocabulary

A polynomial is a sum of terms, each a number times a variable raised to a non-negative whole-number power. Its degree is the highest power, and the leading coefficient multiplies that term: 4x cubed minus 2x plus seven has degree three and leading coefficient four. Order the terms by power first: in five plus 2x squared, the leading coefficient is two.

Adding and subtracting.

ACT 0702 Combining Like Terms

Like terms have the same variables raised to the same powers, and polynomials add or subtract by combining them. Three x squared plus 5x minus two, minus the quantity x squared minus 4x plus six, is 2x squared plus 9x minus eight. Subtracting a polynomial changes the sign of every term in it, not only the first.

Multiplying polynomials.

ACT 0703 Multiplying Polynomials

To multiply polynomials, multiply every term of the first by every term of the second, then combine like terms. For two binomials, FOIL names the four products: first, outer, inner, last. X plus three, times x minus five, is x squared minus 2x minus fifteen. FOIL covers two binomials only.

Squaring a binomial.

ACT 0704 The Square of a Binomial

The square of a binomial is a squared, plus 2 a b, plus b squared, with a minus sign in the middle for a difference: x plus four, squared, is x squared plus 8x plus sixteen. It is not a squared plus b squared: two plus three, squared, is twenty-five, but four plus nine is thirteen.

A sum times a difference.

ACT 0705 Sum Times Difference

A sum times the difference of the same two terms gives the difference of their squares, because the middle terms cancel: x plus seven, times x minus seven, is x squared minus forty-nine. The two terms must match exactly; x plus seven times x minus six does not qualify.

Expanding any product, step by step.

ACT 0706 Expanding a Product

Check for a special pattern first; otherwise multiply every term by every term, keeping each sign with its term, collect like terms, and check by substituting a small number. Two x minus three, times x squared plus 4x minus one, expands to 2x cubed plus 5x squared minus 14x plus three. At x equals one, both sides give negative four. One matching value does not prove the expansion; a second, at x equals two, where both give eleven, makes the check stronger.

Eight entries on factoring, from the common factor to factoring completely.

The first move in factoring.

ACT 0707 The Greatest Common Factor First

Factoring starts with the greatest common factor: the largest number and the highest power of each variable that divides every term. Six x cubed minus 9x squared plus 12x is 3x times the quantity 2x squared minus 3x plus four. Check by distributing back; a common factor left inside makes every later step harder.

A simple trinomial.

ACT 0708 Factoring x² + bx + c

To factor x squared plus b x plus c, find two numbers with product c and sum b. For x squared minus x minus twelve, negative four and three multiply to negative twelve and add to negative one, so x minus four, times x plus three. A positive c means the numbers share a sign; a negative c, opposite signs.

Factoring by grouping.

ACT 0709 Factoring by Grouping

When the leading coefficient is not one, find two numbers with product a c and sum b, split the middle term, and factor in pairs. For 2x squared plus 7x plus three, six and one split it, giving 2x plus one, times x plus three. Pairs that share no binomial signal an error.

A leading coefficient, step by step.

ACT 0710 Factoring a Trinomial With a Leading Coefficient

Take out any common factor, multiply a by c, find the pair of factors that sums to b, rewrite the middle term, then factor each half and the shared binomial. For 6x squared minus 7x minus three, a c is negative eighteen; negative nine and two sum to negative seven; and the result is 3x plus one, times 2x minus three, which multiplies back correctly. When the second pair's common factor is negative, take the sign out with it, so both pairs show the same binomial.

A difference of squares, factored.

ACT 0711 Factoring a Difference of Squares

A difference of two squares factors as a plus b, times a minus b: x squared minus eighty-one is x plus nine, times x minus nine. A sum of two squares cannot be factored. Look for squares hiding behind coefficients: 16x squared is 4x, squared.

Trinomials that are perfect squares.

ACT 0712 Perfect-Square Trinomials

A perfect-square trinomial has square first and last terms and a middle term twice their product, so it factors as a binomial squared. X squared minus 10x plus twenty-five is x minus five, squared, since two times x times five is 10x. The sign inside matches the middle term's sign.

Cubes.

ACT 0713 Sums and Differences of Cubes

A sum or difference of cubes factors into a binomial and a trinomial: x cubed minus eight is x minus two, times x squared plus 2x plus four. The binomial carries the original sign, and the trinomial's middle term carries the opposite sign.

Factoring completely, step by step.

ACT 0714 Factoring Completely

Take out the greatest common factor, count the remaining terms to choose a method, factor, and keep inspecting each new factor. Three x to the fourth minus forty-eight becomes three times x to the fourth minus sixteen, then three times x squared plus four, times x squared minus four, and finally three times x squared plus four, times x plus two, times x minus two. X squared plus four is a sum of squares and stops. Stopping one step early is the error to check for.

Eight entries on rational expressions, and on recognizing equivalent expressions.

Simplifying a fraction of polynomials.

ACT 0715 Cancel Factors, Not Terms

A rational expression is simplified by factoring the numerator and denominator and cancelling common factors. X squared minus nine, over x plus three, is x minus three. Only factors cancel, never terms joined by addition: x plus two over x plus five does not reduce.

Values an expression cannot take.

ACT 0716 Values That Make an Expression Undefined

A rational expression is undefined wherever its denominator equals zero, so those values are excluded: x minus one, over x squared minus four, is undefined at two and negative two. Find them from the original denominator, before cancelling, because cancelling can hide one.

Simplifying, step by step.

ACT 0717 Simplifying a Rational Expression

Factor completely, record every value that makes the original denominator zero, cancel common factors, and write the result with its excluded values. X squared plus 2x minus fifteen, over x squared minus nine, factors to x plus five, times x minus three, over x plus three, times x minus three; x cannot be three or negative three; and the result is x plus five over x plus three. Watch for factors that are negatives of each other: three minus x over x minus three is negative one.

Multiplying and dividing.

ACT 0718 Multiplying and Dividing Rational Expressions

To multiply rational expressions, factor, cancel, and multiply numerators and denominators. To divide, multiply by the reciprocal of the divisor: x over x plus one, divided by x squared over x plus one, is one over x. Flip only the second expression.

Adding rational expressions.

ACT 0719 Adding Rational Expressions

To add or subtract rational expressions, rewrite them over the least common denominator, then combine the numerators: two over x plus three over x plus one is 5x plus two over x times x plus one. Never add denominators: one half plus one third is five sixths, not two fifths.

Adding two expressions, step by step.

ACT 0720 Adding Two Rational Expressions

Factor each denominator, build the least common denominator, multiply each numerator by the factors its denominator lacks, then combine and simplify. Three over x squared minus four, minus one over x minus two, becomes three minus the quantity x plus two, over x plus two, times x minus two, which is one minus x over that denominator. Subtracting a numerator subtracts every term: writing three minus x plus two gives five minus x, the slip to watch for.

Fractions inside fractions.

ACT 0721 Simplifying a Complex Fraction

A complex fraction is simplified by finding the least common denominator of every small fraction, multiplying the top and bottom of the large fraction by it, and simplifying. One plus one over x, divided by one minus one over x squared, multiplied through by x squared, becomes x over x minus one. Multiply every term by the common denominator, including whole numbers such as one.

Recognizing an equivalent expression.

ACT 0722 Recognizing an Equivalent Expression

Simplify the given expression, simplify or compare each choice, and if two look close, substitute a number into the original and each choice. X plus three, squared, minus x minus three, squared, expands to 12x, and at x equals two both give twenty-four, while 2x squared plus eighteen gives twenty-six. One mismatch rules a choice out, but one match proves little; avoid testing with zero or one, where many expressions agree.

Look at what you can now do. You can add, subtract and multiply polynomials, changing every sign when you subtract. You can expand with the special patterns and check with two values. You can factor completely: common factor first, then the method the terms suggest, until nothing factors. You can simplify and combine rational expressions, cancelling factors and keeping the excluded values. And you can recognize an equivalent expression, testing with numbers other than zero and one. Every rewrite kept the value the same. Unit nine turns to lines and linear equations.