GMAT Exam: A Simple Guide To Algebra - Polynomials

Basic Concepts
In an algebraic expression, terms are the parts of the expression that are connected to the other parts by plus or minus signs.

For example, the expression has three terms:

–8x2, , and 27.

If the algebraic expression has no plus or minus signs, then the algebraic expression itself is a term.

A monomial is a term that when simplified is a constant or a product of one or more variables raised to positive integer powers, with or without an explicit coefficient.

In monomials, no variable divisors, negative exponents, or fractional exponents are allowed.

For example, 20, 5x, –54x, , 6x2, and –100x2y2 are monomials; but , –7x–3, and are not monomials.

Constants like 20 are monomials because you can write 20 as 20(1) = 20x0.

A polynomial is a single monomial or a sum of monomials. A polynomial that has exactly one term is a monomial. A polynomial that has exactly two terms is a binomial. A polynomial that has exactly three terms is a trinomial. A polynomial that has more than three terms is just a general polynomial.

Monomials that are constants or that have exactly the same variable factors (that is, the same letters with the same corresponding exponents) are like terms. Like terms are the same except, perhaps, for their coefficients. For example, –10x and 25x are like terms; 4xy2 and 5xy2 are like terms; but 4xy2 and 5x2y are not like terms. Monomials that are not like terms are unlike terms.

Addition and Subtraction of Monomials
- Adding Monomials That Are Like Terms: Add their numerical coefficients and use the sum as the coefficient of their common variable component. For example, –10x + 25x = 15x.
- Subtracting Monomials That Are Like Terms: Subtract their numerical coefficients and use the difference as the coefficient of their common variable component. For example, 4xy2 – 9xy2 = –5xy2.
- Adding or Subtracting Unlike Terms: Simply indicate the addition or subtraction. For example, 4xy2 + 5x2y is 4xy2 + 5x2y. (These two terms cannot be combined into a single monomial term.)

Simplifying Polynomials
Polynomials are in simplest form when they contain no uncombined like terms. To organize the process of simplifying a polynomial, use the properties of real numbers to rearrange the expression so that matching like terms are together (later, you might choose to do this step mentally). If the expression includes unlike terms, just indicate the sums or differences of such terms.

Tip: To avoid sign errors, change instances of – – to + and change instances of – + or + – to –. Thereafter, keep a – sign with the number that follows it.

Here is an example.


As shown in this example, polynomials in one variable are usually written in descending order of the variable’s exponents.

Dealing with Parentheses
If no sign or if a plus (+) sign immediately precedes parentheses that enclose an algebraic expression, remove the parentheses and rewrite the algebraic expression without changing any signs.

Here is an example.


If a negative (–) sign immediately precedes parentheses that enclose an algebraic expression, remove the parentheses and the negative sign and rewrite the algebraic expression but with all the signs changed.

Tip: Change all the signs, not just the first one.

Here is an example.
–(–7x2 – 5x + 3) = 7x2 + 5x – 3

If a minus (–) sign immediately precedes parentheses that enclose an algebraic expression, remove the parentheses and the minus sign and rewrite the algebraic expression but with all the signs changed.

Here is an example.

As you can see, negative and minus signs produce similar results.

However, keep in mind that a negative sign is used to indicate the opposite of a number, but a minus sign is used to indicate subtraction.

If a number immediately precedes parentheses that enclose an algebraic expression, multiply the coefficient of each term inside the parentheses by the number.

Tip: Multiply each coefficient, not just the first one.

Here are examples.
2(4x3 – 5x2 – x + 3) = 8x3 – 10x2 – 2x + 6 and –2(4x3 – 5x2 – x + 3) = –8x3 + 10x2 + 2x – 6

Multiplying Polynomials

- To Multiply Monomials: Multiply the numerical coefficients, multiply the variable factors using rules for exponents, and then use the product of the numerical coefficients as the coefficient of the product of the variable factors to obtain the answer.

Here are examples. (5x)(–2x) = –10x2, (–5x3)(–2x4) = 10x7, and (x3y4)(xy2) = x4y6
Remember, x = x1.

- To Multiply a Polynomial by a Monomial: Multiply each term of the polynomial by the monomial. Here are examples.
x(3x – 2) = 3x2 – 2x, 5x(2x + 4) = 10x2 + 20x, and
–8x3y4(2x2 – 7xy2 – 3) = –16x5y4 + 56x4y6 + 24x3y4.

- To Multiply Two Binomials: Multiply all the terms of the second binomial by each term of the first binomial and then simplify. The FOIL method is a quick way to perform the multiplication.

The word FOIL is an acronym for first, outer, inner, and last.

Here is an example.


The inner and outer products are the middle terms.
- To Multiply Two General Polynomials: Multiply all the terms of the second polynomial by each term of the first polynomial and then simplify.

Here is an example.


Special Products
Here is a list of the forms of special products of polynomials that are helpful to know.

- Perfect Squares (x + y)2 = x2 + 2xy + y2; for example, (x + 3)2 = x2 + 6x + 9 (x – y)2 = x2 – 2xy + y2; for example, (x – 3)2 = x2 – 6x + 9

- Difference of Two Squares (x + y)(x – y) = x2 – y2; for example, (x + 3)(x – 3)2 = x2 – 9
Notice that the two middle terms sum to zero.

- Perfect Cubes


- Sum of Two Cubes (x + y)(x2 – xy+y2) = x3 + y3; for example, (x + 1)(x2 – x +1) = x3 + 1

- Difference of Two Cubes (x – y)(x2 + xy+y2) = x3 – y3; for example, (x – 1)(x2 + x +1) = x3 – 1

You can substitute any value or variable expression for x and y in the above forms.

Here are examples.


Factoring Polynomials
Factoring is the process of undoing multiplication. Here are some general guidelines for factoring polynomials.

- If the expression has a greatest common factor (GCF), factor out the GCF. The GCF is the product of the greatest common numerical factor and a second component made up of the common variable factors, each with the highest power common to each term.

Here is an example.
15x3y4 – 3x2y3 + 12xy2 = 3xy2 (5x2y2 – xy + 4)

- If there are two terms, check for a special binomial product.

1. Difference of two squares: (4x2 – 9) = (2x + 3)(2x – 3)
Expressions that have the form (x2 + y2) are not factorable on the GMAT. For example, (4x2 + 9) is not factorable.

2. Difference of two cubes: 8x3 – 27 = (2x – 3)(4 x2 + 6x + 9)

3. Sum of two cubes: 64x3 +125 = (4x + 5)(l6x2 – 20x + 25)

- If there are three terms, check for a quadratic trinomial. A quadratic trinomial has the form ax2 + bx + c, where a, b, and c are nonzero constants. You can use trial and error to factor simple quadratic trinomials.

As you proceed, call to mind the FOIL method for multiplying two binomials. When you use FOIL to multiply (x + k)(x + h), where k and h are constants, you get x2 + (k + h)x + kh. If the coefficient of the middle term, (k + h), is not zero, the product is a quadratic trinomial.

If k and h have the same sign, the last term of the trinomial is their product, which is positive, and the coefficient of the middle term is their algebraic sum, with the same sign.

If k and h have different signs, the last term of the trinomial is their product, which is negative, and the coefficient of the middle term is their algebraic sum, with the sign of whichever one has the greater absolute value.

Here are examples.


Your task when factoring is to reverse the above process.

Here are examples.
4. To factor x2 – 9x + 14, observe that the last term is positive and the middle term is negative. So, you want to find two negative numbers whose product is 14 and whose sum is –9. Thus, x2 – 9x + 14 = (x – 7)(x – 2).

5. To factor x2 – 5x – 14, observe that the last term is negative and the middle term is negative. So, you want to find two numbers with different signs whose product is –14 and whose sum is –5. Thus, x2 – 5x – 14 = (x – 7)(x + 2).

Always check whether any previously obtained factor can be factored further.

For example,