Mathe: Algebraic Reasoning

polynomial is in the form x2+2xy+y2x2+2xy+y2, what is its factored form?

The correct answer is: (x+y)(x+y)=(x+y)2(x+y)(x+y)=(x+y)2 Explanation: On the other hand, the factored form of: x2−2xy+y2x2−2xy+y2 is (x−y)(x−y)=(x−y)2

What is the difference between a function and a relation?

The correct answer is: A function is a specific form of relation where every value of xx has only one value of yy that makes the relation true. Explanation: y=2xy=2x is a function because for each value of xx there is only one value that equals 2x2x. y<2xy<2x is a relation because for each xx, there are many values that are less than 2x2x.

Which of the following is a polynomial? A. 3x63x6 B. 3x2+4x−103x2+4x−10 C. 2x2⋅−3x

The correct answer is: B Explanation: A monomial is an expression with one term. A polynomial is an expression of different monomials added to, or subtracted from, each other. Choice A is a monomial. Choice C is two monomials multiplied together, resulting in another monomial (2x2⋅−3x=−6x32x2⋅−3x=−6x3), not a polynomial. Choice B consists of three monomials added/subtracted to/from each other. Thus, this is a polynomial.

Which of the following is not a radical expression? A. 2x−−√2x B. 4-√×5-√4×5 C. 3x3−4x

The correct answer is: C Explanation: A radical expression is an expression that contains roots. It may contain numbers and/or variables. Choice A is a radical expression since it has both a number and a variable under a root. Choice B is also a radical expression as it contains roots. Choice C doesn't have any roots. It is not a radical expression, but rather a polynomial.

Which of the following is an exponential function? A. y=3x2−2y=3x2−2 B. y=4x−−√+2y=4x+2 C. y=2x+4

The correct answer is: C Explanation: An exponential function is a mathematical function where a constant is raised to a variable power. It is of the form y=axy=ax where aa is a constant and xx is a variable. From the choices given, C is the only exponential function.

What is the next step in simplifying the polynomial shown below? 3x−6+22−6x+4

The correct answer is: Evaluate the exponent: 22=422=4 Explanation: Remember to follow the order of operations (PEMDAS):Do everything in parentheses (P), left to right.Evaluate any exponents (E), left to right.Do all multiplication and division (MD), in order, left to right.Then do all addition and subtraction (AS), in order, from left to right. A good way to remember: Please Excuse My Dear Aunt Sally. Since there aren't any parentheses in the expression, the next order of operation to deal with is evaluating exponents. Thus, the next step is to evaluate 2222.

The exponential model P=1000(1.3)5P=1000(1.3)5 shows the population of a city after a certain number of years. What does 1.31.3 represent?

The correct answer is: It represents the growth factor of the population with a 30%30% growth rate per year. Explanation: The exponential growth model is given by the equation: a=a0(1+r)ta=a0(1+r)t Where aa is the final amount, a0a0 is the initial amount, rr is the growth rate, and tt is the time period. Looking at our equation P=1000(1.3)5P=1000(1.3)5, we can clearly see that 1.31.3 is the growth factor. Solving for rr: 1.3=1+r1.3=1+r r=1.3−1r=1.3−1 r=0.3r=0.3 In percent, that is 0.3×100=30%0.3×100=30%. So, 1.31.3 represents the growth factor with a growth rate of 30%30%.

A student's first three steps to solve a radical equation are shown below. In which step does the student make his first mistake? Equation: x+5−−−−−√=3x+5=3 Step 1: (x+5−−−−−√)2=(3)2(x+5)2=(3)2Step 2: (x+5)2=9(x+5)2=9Step 3: x+5=3

The correct answer is: Step 2 Explanation: The first step in solving a radical equation is to square both sides in order to eliminate the square root sign. Step 1 is correctly done. In step 2, the student does get rid of the square root sign but keeps the square. This is incorrect. The squaring would eliminate the square root sign leaving only x+5x+5, not (x+5)2(x+5)2.

What is meant by the "degree of a polynomial"?

The correct answer is: The degree of a polynomial is the highest power of a variable in that polynomial. Explanation: The degree of the polynomial 4x2+2x34x2+2x3 is 33.

What is the vertex of a parabola?

The correct answer is: The vertex is the point where a parabola changes direction, or, the highest/lowest point in the graph of a quadratic. Explanation: The vertex of a parabola is the point where the graph of a parabola changes direction. It is the lowest point of a positive parabola (opens upward) and the highest point of a negative parabola (opens downward).

What is the general form of a quadratic equation?

The correct answer is: ax2+bx+c=0ax2+bx+c=0 Explanation: The general form of a quadratic equation is ax2+bx+c=0ax2+bx+c=0. The coefficient of x2x2 is aa, the coefficient of xx is bb, and the constant term is cc.

True or false? The graph of the quadratic function −x2+2x+5−x2+2x+5 opens upward.

The correct answer is: false Explanation: The general form of a quadratic function is ax2+bx+cax2+bx+c. If a>0a>0, the graph opens upward. If a<0a<0, the graph opens downward. For the function −x2+2x+5−x2+2x+5, the value of aa is −1−1. Since a<0a<0, the graph will open downward.

True or false? A system of linear equations has one solution (x,y)(x,y) that works in both equations.

The correct answer is: false Explanation: The lines might be parallel, in which case there are no solutions, or the equations might be 22 different forms of the same line, such as x=yx=y and 2x=2y2x=2y. In this case, there are infinite solutions.

In a linear equation of the form y=mx+by=mx+b, what does mm represent?

The correct answer is: mm represents the slope of the line. Explanation: The slope is how much the yy value changes for a unit change in xx. The slope is calculated by dividing the change in yy values by the change in xx values for two points on the line.

What are the three different solution sets of a system of linear equations?

The correct answer is: no solution, unique solution, and infinite solutions Explanation: When two lines are parallel to each other, the system has no solution and there is no intersection point: y=2x+4y=2x+4 and y=2x+10y=2x+10 When two lines intersect at one point, the intersecting point is the unique solution of the system: y−2x=5y−2x=5 and 3x−y=53x−y=5 When two equations of a linear system are essentially the same line (constant multiple of each other), we have infinitely many solutions: −4x+y=2−4x+y=2 and −12x+3y=6

To evaluate the function f(x)=x2−2x+5f(x)=x2−2x+5 at x=−2x=−2, we will ____.

The correct answer is: substitute −2−2 into all xx of the function f(x)f(x) and simplify to get the answer Explanation: When we want to evaluate a function f(x)f(x) at a value x=ax=a, we simply plug in aa into the function. This means substituting aa into all xx of the function given and arriving at an answer. The evaluation of the function f(x)f(x) at x=−2x=−2 is shown below. f(x)=x2−2x+5f(x)=x2−2x+5 f(−2)=(−2)2−2(−2)+5f(−2)=(−2)2−2(−2)+5 f(−2)=4+4+5f(−2)=4+4+5 f(−2)=13

True or false? In an inequality, multiplying or dividing by a negative number reverses the inequality sign.

The correct answer is: true Explanation: We follow the equation-solving rules when dealing with inequalities. The only exception is when we multiply or divide by a negative number, we reverse the inequality sign. −2x+6>10−2x+6>10 −2x>10−6−2x>10−6 −2x>4−2x>4 −2x−2>4−2−2x−2>4−2 x<−2x<−2 Note how we changed the inequality sign from >> to << in the last step when dividing by a negative number.

Simplify the following expression using exponent rules. (2xyx−2)3

he correct answer is: 8x9y38x9y3 Explanation: Let's simplify the expression inside the parentheses: (2xyx−2)3(2xyx−2)3 =(2xyx2)3=(2xyx2)3 =(2x3y)3=(2x3y)3 Now, we can raise the expression inside the parentheses to the third power. (2x3y)3(2x3y)3 =23(x3)3y3=23(x3)3y3 =8x9y3

For the rational function y=3x+5x+1y=3x+5x+1, what happens to the graph at x=−1x=−1?

he correct answer is: There is a vertical asymptote at x=−1x=−1. Explanation: Rational functions have a vertical asymptote at the point where the denominator of the function equals 00. For the function y=3x+5x+1y=3x+5x+1: x+1=0x+1=0 x=−1x=−1 There is a vertical asymptote at x=−1x=−1 since −1−1 makes the function value undefined.

Describe the nature of roots of a parabola with a discriminant equal to 0.

he correct answer is: one real root (zero) Explanation: The quantity b2−4acb2−4ac is known as the discriminant and it helps us find the nature of roots of a quadratic function. If the discriminant <0<0, there are no real roots of the quadratic. If the discriminant =0=0, there is one real root of the quadratic. If the discriminant >0>0, there are two real roots of the quadratic.