Algebra and Geometry Review – Exercise 08

LESSON 1: Degree and leading coefficient of a univariate polynomial

What are the leading coefficient and degree of the polynomial?
3-3x^2+6x

EXPLANATION: We first rewrite this polynomial in standard form. 
We rearrange the terms so that the exponents on the variable decrease from left to right.
3-3x^2+6x=-3x^2+6x+3 standard form
The leading term is the first term when the polynomial is in standard form.
So, for this polynomial, the leading term is -3x^2

Leading coefficient
The coefficient of a term is the number multiplying the variable,
when there is no variable, the coefficient is the term itself
The leading coefficient is the coefficient of the leading term.
For our polynomial, the leading term is -3x^2. So, the leading coefficient is -3.

Degree
The degree of a term is the exponent of its variable,
when there is no variable, the degree is zero
The degree of a polynomial is the degree of its leading term
For our polynomial, the leading term is -3x^2. So, the degree of the polynomial is 2.


ANSWER: Leading coefficient -3, Degree 2

LESSON 2: Multiplying binomials in two variables

Multiply and Simplify your answer (5x+3z)(8x-z)

EXPLANATION: 
We can multiply 5x+3z by 8x-z using the FOIL (First, Outer, Inner, Last) method.
F: Multiply the two First terms 5x*8x=40x^2
O: Multiply the two Outside terms 5x*(-z)=-5xz
I: Multiply the two Inside terms 3z*8x=24xz
L: Multiply the two Last terms 3z*(-z)=-3z^2
The product  (5x+3z)(8x-z) is then equal to the sum of these terms.
(5x+3z)(8x-z)=40x^2-5xz+24xz-3z^2
=40x^2+19xz-3z^2

ANSWER: 40x^2+19xz-3z^2

LESSON 3: Restriction on a variable in a denominator (Linear)

Find all excluded values for the expression.
That is, find all values of w for which the expression is undefined.
(11w+12)/(w+3)
If there is more than one value, separate them with commas.

EXPLANATION: Division by zero is not defined, so the expression is undefined when its denominator is zero.
We must find all values of w for which the expression is undefined.
So we set the denominator equal to 0 and solve for w.
w+3=0
w=-3
Thus, (11w+12)/(w+3)  is undefined when w=-3

ANSWER: w=-3

LESSON 4: Square root multiplication (Basic)

Simplify √8*√6

EXPLANATION: We'll use the following property of square roots to simplify our expression.
Product property of square roots:
√a*√b=√ab for any nonnegative numbers a and b
We simplify as follows:
√8*√6= √(8*6) Using the product property of square roots
=√48 Multiplying under the square root sign
=√(16*3) Factoring out the perfect square 16
=√16*√3 Using the product property of square roots
=4√3
ANSWER: 4√3

LESSON 5: Signed fraction division

Divide and write your answer as a fraction or mixed number in simplest form.
-10/21÷(-15/7)

EXPLANATION: Dividing by -15/7 is the same as multiplying by its reciprocal -7/15
-10/12  ÷(-15/7)=-10/21*-7/15
=-(10 2)/21* -7/(15 3) Canceling (dividing by) the common factor 5
=-(10 2)/(21 3)* -(7 1)/(15 3) Canceling (dividing by) the common factor 7
=2/9 Note that the answer is positive because we're multiplying two negative numbers.

ANSWER: 2/9