Algebra• Seeing Structure in

Expressions

Arithmetic with Polynomials and Rational Expressions

• Creating Equations

• Reasoning with Equations and Inequalities

Arithmetic with Polynomials and Rational Expressions

Perform arithmetic operations on polynomials.

A-APR.1Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Algebra I & II

Arithmetic with Polynomials and Rational Expressions

Perform arithmetic operations on polynomials.

What type of skills do students need in order to add, subtract, and multiply polynomials?

Arithmetic with Polynomials and Rational Expressions

Understand the relationship between zeros and factors of polynomials.

A-APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x−a is p(a), so p(a)=0 if and only if (x−a) is a factor of p(x).  

Algebra II

Arithmetic with Polynomials and Rational Expressions

Applying the Remainder TheoremIs x – 4 a factor of the polynomial p(x) = 5x3 – 13x2 – 30x + 8?Rather than divide p(x) by x – 4, apply the Remainder Theorem as a shortcut:If p(4) = 0, then we know that x – 4 is a factor.p(4) = 5(4)3 – 13(4)2 – 30(4) + 8p(4) = 5(64)-13(16)-120 + 8p(4) = 0So x – 4 is a factor of p(x)

Arithmetic with Polynomials and Rational Expressions

Understand the relationship between zeros and factors of polynomials.

The zero-factor principle states that

If A*B = 0, then either A = 0, or B = 0 (or both)

Arithmetic with Polynomials and Rational Expressions

Understand the relationship between zeros and factors of polynomials.

A-APR.3

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Algebra II

Arithmetic with Polynomials and Rational Expressions

Understand the relationship between zeros and factors of polynomials.

Example:Find a polynomial that has degree 4 and has zeros att = -1, t = 0, t = 1, and t = 2

Arithmetic with Polynomials and Rational Expressions

Use polynomial identities to solve problems.

A-APR.4

Prove polynomial identities and use them to describe numerical relationships.

Algebra II

Arithmetic with Polynomials and Rational Expressions

Using polynomial identities:x2 – y2 = (x + y)(x – y)so16 – 25 = (4 + 5)(4 - 5)(7 – 3)(7 + 3) = 49 – 9

Show the area model for (x + y)2

Where does the 2xy appear?

(105)2 = (100 + 5)2

= 1002 + 2(5)(100) + 52

= 10000 + 1000 + 25 = 11025

Arithmetic with Polynomials and Rational Expressions

Use polynomial identities to solve problems.

A-APR.5

(+) Know and apply the Binomial Theorem for the expansion of (x+y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.*

Algebra II (+) Honors or Year 4

Arithmetic with Polynomials and Rational Expressions

Rewrite rational expressions. A-APR.6

Rewrite simple rational expressions in different forms;

write a(x)b(x) in the form q(x)+r(x)b(x), where

a(x), b(x), q(x), and r(x) are polynomials with the degree

of r(x) less than the degree of b(x),

using inspection, long division, or, for the more complicated examples, a computer algebra system.

Algebra II

Arithmetic with Polynomials and Rational Expressions

Use long division to determine the quotient of (x3 -1)/(x2+1)

Express the fraction in quotient form.

How does this quotient relate to the graph of f(x) = (x3 -1)/(x2+1) ?

Arithmetic with Polynomials and Rational Expressions

Rewrite rational expressions.

A-APR.7(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Algebra II (+) Honors or Year 4

Algebra• Seeing Structure in

Expressions

• Arithmetic with Polynomials and Rational Expressions

Creating Equations

• Reasoning with Equations and Inequalities

Creating Equations

Create equations that describe numbers or relationships.

A-CED.1Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. ⋆

Algebra I & II

Creating Equations

Suppose a friend tells you she paid a total of $16,368 for a car, and you'd like to know the car's list price (the price before taxes) so that you can compare prices at various dealers. Find the list price of the car if your friend bought the car in:Arizona, where the sales taxis 6.6%.New York, where the sales tax is 8.25%.A state where the sales tax is r.

Creating Equations

Michelle will get a final course grade of B+ if the average on four exams is greater than or equal to 85 but less than 90. Her first three exam grades were 98, 74, and 89. What fourth exam grade will result in a B+ for the course?

Creating Equations

Create equations that describe numbers or relationships.

A-CED.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. ⋆

Algebra I & II

Creating Equations

Observations show that the heart mass H of a mammal is 0.6% of the body mass M, and that the blood mass B is 5% of the body mass.(a) Write a formula for M in

terms of H(b) Write a formula for M in

terms of B(c) Write a formula for B in

terms of H. Is this consistent with the statement that the mass of blood in a mammal is about 8 times the mass of the heart?

Creating Equations

A borehole is a hole dug deep in the earth for oil or mineral exploration. Often temperature gets warmer at greater depths. Suppose that the temperature in a borehole at the surface is 4 oC and rises by 0.02oC with each additional meter of depth. Express the temperature T in oC in terms of depth d in meters. Graph the equation on a set of axes. What does the T-intercept represent in terms of the problem? If the temperature is 24oC, how deep is the hole?

Creating Equations

Create equations that describe numbers or relationships. A-CED.3

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. ⋆

Algebra I & II

Creating Equations

A newly designed motel has S small rooms measuring 250 ft2 and L large rooms measuring 400 ft2 of available space. The designers have 10,000 ft2 of available space. Write an equation relating S and L.

Creating Equations

Create equations that describe numbers or relationships.

A-CED.4

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V=IR to highlight resistance R. ⋆

Algebra I & II