Equations of lines
[5+(2b+2)]^2
(A+B)^2=A^2+2AB+B^2 B2 = (2b+2)^2 = 4b^2+8b+4 2AB=(2)(5)(2b+2) = 20b+20 A^2=5*5 = 25 solution: 4b^2+28b+49
The formulas for the square of a binomial are below.
(a+b)2=a2+2ab+b2, (a−b)2=a2−2ab+b
Multiplication neg/pos rules
+ x + = + - x - = + - x + = - + x - = -
Slopes of perpendicular lines have a product of?
-1
can a polynomial function be evaluated only for a numerical value?
A variable can be replaced by an equivalent variable expression. For example, the polynomial function f(x)=x2+2x can be evaluated for x=a+b by substituting a+b for x. This gives f(a+b)=(a+b)2+2(a+b).
Finding the value of a polynomial function at a given replacement value is similar to what?
Evaluating a polynomial for a given replacement value Finding the value of a polynomial function for a given replacement value is similar to evaluating a polynomial for a given replacement value. In each case, replace x with the given replacement value and simplify using the order of operations.
Name at least one other method that can be used to muliply polynomials. (6x+1)^2
FOIL order of distributive order The expression (6x+1)2 can be rewritten as (6x+1)(6x+1). Then it can be multiplied using FOIL order or the distributive property.
What type of multiplication of polynomials uses FOIL?
FOIL stands for "First-Outer-Inner-Last." This is an abbreviation for four terms. When multiplying a binomial by a binomial, the result is four terms (before combining like terms). They are the first terms multiplied, the outer terms multiplied, the inner terms multiplied and the last terms multiplied. As an example, (4x+1)(3x+7)=12x2+28x+3x+7.
Slopes of Parallel Lines
In a coordinate plane, two non-vertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.
How can a polynomial subtraction problem be changed into an equivalent addition problem?
Recall that subtraction is equivalent to adding the opposite. If F(x) and G(x) are polynomials, then F(x)−G(x)=F(x)+(−G(x)).
Why is the degree of each term found in a polynomial?
The degree of the polynomial is the greatest degree of any of its terms. Therefore, the degree of each term must be calculated before the degree of the polynomial can be determined.
What property is key when multiplying polynomials?
The distributive property is key. The product rule for exponents is most often used when applying the property.
Multiply. (x+7)(x+7)(x^2+49)
The formulas below are used to find the product of a sum and difference of two terms and the square of a binomial. (a+b)(a−b)=a^2−b^2 (a+b)^2=a^2+2ab+b^2 (a−b)^2=a^2−2ab+b^2
Find an equation of the line. Write the equation using function notation. Through (−3,−3); parallel to x+3y=4
Two nonvertical lines are parallel if they have the same slope and different y-intercepts. 1: find the slope of the given line using y=mx+b 3y=4-x dived by 3y y=-1/3x+4/3 m=-1/3 slopes of parallel lines are equal -1/3 writes in point slope form: -1/3 slope (-3,3) y-(-3) = -1/3[x-(-3)] y+3=-1/3 x-1 y=-1/3 x-4 solution: f(x)= -1/3 x - 4
Find the equation of each line. Through (20,16); perpendicular to y=−19.
Two nonvertical lines are perpendicular if the product of their slopes is −1. In other words, two nonvertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other. Notice that the equation of the given line is of the form y=c. Therefore, the given line y=−19 is a horizontal line. A line perpendicular to a horizontal line is a vertical line. Recall that the equation of a vertical line has the form x=c. The constant, c, is the x-coordinate for all points on the line. The x-coordinate of the given point (20,16) is 20. Therefore, the equation of the line that is perpendicular to y=−19 and goes through the point (20,16) solution is x=20.
Find an equation of the line having the given slope and containing the given point. Slope 10; through (5,12)
Use point-slope equation of a linear equation y−y1=m(x−x1). Substitute known values y−12=10(x−5) -solve solution written slope-intercept form y=mx + b (y=10x-38)
why does multiplying the sum and difference of the same two terms always give a binomial answer?
Using the FOIL order technique, (a+b)(a−b)=a2−ab+ab−b2. The product of the outside terms will always be −ab, and the product of the inside terms will always be ab, so they will always be opposites and subtract out. This leaves only the product of the first terms and the product of the last terms, a2−b2.
What is a polynomial?
a term in algebra that uses one or more variable.
A fruit company recently released a new applesauce. By the end of its first year, profits on this product amounted to $15,000. The anticipated profit for the end of the fourth year is $24,000. After the first year, the ratio of change in time to change in profit is constant. Let x be years and P be profit in dollars.
a. Write a linear function P(x) that expresses profit as a function of time. When x=1, y=15,000. When x=4, y=24,000. Write a linear equation that passes through the points (1,15000) and (4,24000). First, find the slope using the equation m=y1−y2x1−x2. m=24,000−15,0004−1=3,000 Recall that the point-slope equation for the line is y−y1=mx−x1, where m is the slope of the line and x1,y1 is a point on the line. Using the slope 3,000 and the point (1,15000), write the equation in point-slope form. y−y1=mx−x1 y−15,000=3,000(x−1) Let m=3,000 and x1,y1=(1,15000). y=3,000x+12,000 Solve for y. P(x)=3,000x+12,000 Write the equation as a function. Thus, the linear function P(x)=3,000x+12,000 expresses profit as a function of time. b. Use this function to predict the company's profit at the end of the tenth year. Substitute x=10 into the function and simplify. P(x)=3,000x+12,000 P(10)=3,000(10)+12,000 Substitute x=10. P(10)=$42,000 Simplify. Thus, using this function to predict, the company's profit at the end of 10 years will be $42,000. c. Predict when the profit should reach $102,000. Set P(x) equal to the target profit level of $102,000, and solve for x. P(x)=3,000x+12,000 102,000=3,000x+12,000 Substitute P(x)=102,000. 102,000−12,0003,000=x Isolate x. 30=xCalculate x. Thus, the profit should reach $102,000 in 30 years.
function notation
an equation in the form of 'f(x)=' to show the output value of a function, f, for an input value x
A _________ is a polynomial with exactly two terms.
binomial A monomial is a polynomial with exactly one term, a binomial is a polynomial with exactly two terms, and a trinomial is a polynomial with exactly three terms.
The numerical factor of a term is the __________.
coefficient A term is a number or the product of a number and one or more variables raised to powers. The numerical coefficient, or simply the coefficient, is the numerical factor of a term. For example, in the term −1.2x^2, the number −1.2 is the coefficient.
The __________ of a polynomial is the largest degree of all its terms.
degree The degree of a polynomial is the greatest degree of any of its terms. For example, the degree of the polynomial 2x^3+5x^2+7 is 3 because 3 is the greatest degree of any of its terms.
function notation
f(x) = -2x-1
Find the equation of the line. Through (−10,−2) and (0,−7) Which of the following is the equation of the line in standard form?
find slope using the two points m=y2-y1/x2-x1 use y-y1=m(x-x1) y-(-4)=-4/3[x-(-3)] y+4=-4/3(x+3) write equation in standard form Ax+By=C 3y+12=-4(x+3) **multiple by 3 to rid fraction 3y+12=-4x-12 **distributive property 4x+3y+12=-12 **move the variable term to left 4x+3y=-24 **move the constant term to right solution : 4x + 3y = -24
Find an equation of the line. Write the equation in function notation. Through (-10, 8); perpendicular to 2y = x - 6
find the slope-intercept equation by solving for y. divide both sides by 2. 2y=x-6 divide 2y y= 1/2 x - 3 find the slope of the new line. mperp = -1/1/2 or -1*2 mperp = -1 * 2 = -2 use y-y1=m(x-x1) or y - 8= -2(x-(-10)) write in y-mx+b form to solve for y y-8=-2(x+10) simplify y-8 = -2x - 20 add 8 to both sides y = -2x-12
Through (-3,-6); parallel to 5x+3y = 8
setup in y=mx+b ** 5x+3y=8 to 3y=5x-8 divide by 3y y=5/3x-8/3 Parallel coefficent is -1
Find the equation of the line. Through (-10,-2) and (0,-7)
x + 2y = -14
Point slope equation of a line
y - y1 = m(x - x1)
slope-intercept form
y=mx+b, where m is the slope and b is the y-intercept of the line.
(a+b)2 = a2 +2ab+b2 and (a−b)2 = a2 −2ab+b2.
(a+b)2=a2+2ab+b2 and (a−b)2 = a2 −2ab+b2.
