solving systems of equations - ms. griggs · 2019-10-27 · solving systems of equations. bell...

Solving

Systems

of

Equations

Bell Ringer

Solve the following equation by completing the square:

r2 – 10r + 26 = 8

Simplify the following expression:

(–4 – 3i) – (8 – 2i)

Solving

Linear-

Quadratic

Systems

What is a system?

– More than one equation that has to be true at the same time.

– What do solutions to systems of equations look like? (What is the

solution?)

Solving Linear Quadratic

Systems: Algebraically

y = -x2 – 8x – 10

2x + y = -1

Solving Linear-Quadratic


-x2 – 7x – 2y + 6 = 0

3x + y = 0



3x – y = 7

y + 4 = 2(x + 5)2

Solving Linear Quadratic

Systems: Graphically

– Doc Camera


Systems: Homework

–Pg. 197:1-14, 16, 17, 19

Bell Ringer

Solve:

1.𝑥2 + 𝑥 − 𝑦 − 5 = 0

2.𝑥 + 𝑦 + 4 = 0

Vocabulary

– What is a system?

– It is more than one equation that has to be true at the

same time.

– The solution to a system is what?

Vocabulary

Examples

Solve the following system to find the solutions for x, y, and z.

−2𝑥 + 𝑦 + 3𝑧 = 20

−3𝑥 + 2𝑦 + 𝑧 = 21

3𝑥 − 2𝑦 + 3𝑧 = −9

Examples


2𝑥 − 𝑦 − 3𝑧 = 1

4𝑥 + 3𝑦 + 2𝑧 = −4

−3𝑥 + 2𝑦 + 5𝑧 = −3

Examples


3𝑥 + 4𝑦 − 𝑧 = −7

𝑥 − 5𝑦 + 2𝑧 = 19

5𝑥 + 𝑦 − 2𝑧 = 5

Examples


2𝑥 − 𝑦 + 3𝑧 = −12

−𝑥 + 2𝑦 − 3𝑧 = 15

𝑦 + 5𝑧 = −6

Bell Ringer

Polynomials

Adding, Subtracting, & Multiplying

Vocabulary

– What is a polynomial?

– A polynomial is a monomial or a sum of monomials.

– What is a monomial?

– A monomial is a real number, a variable, or a product of a real number and one or more

variable with whole-number exponents.

– What is a binomial?

– A binomial has two terms that are either added or subtracted.

Vocabulary

– What is a trinomial?

– A trinomial has three terms that are either added or subtracted.

– How do you find the degree of a polynomial?

– The degree of a monomial is the sum of the exponents of its variables. The degree of

a polynomial in one variable is the same as the degree of the monomial with the

greatest exponent.

Vocabulary

Adding Polynomials

Simplify each.

1. (5r3 + 8) + (6r3 + 3)

2. (6x2 + 7) + (3x2 + 1)

3. (3z3 – 4z + 7z2) + (8z2 – 6z – 5)

4. (2k2 – k + 3) + (5k2 + 3k – 7)

Adding Polynomials

– 4𝑥2 − 𝑥3 + 2 + 5𝑥4 + −𝑥 + 6𝑥2 + 3𝑥4

– 17𝑥4 + 8𝑥2 − 9𝑥7 + 4 − 2𝑥3 + (11𝑥3 − 8𝑥2 + 12)

Solving Polynomials(Simplify Each)

(x3 – 3x2 + 5x) – (7x3 + 5x2 – 12)

(–4m3 – m + 9) – (–4m2 + m – 12)

(x2 – 2) – (3x + 5)

(14h4 + 3h3) – (9h4 +2h3)

(–9r3 + 2r – 1) – (–5r2 + r + 8)

(y3 – 4y2 – 2) – (6y3 + 4 – 6y2)

Subtracting Polynomials

12𝑥3 + 5𝑥 − 8𝑥2 + 19 − 6𝑥2 − 9𝑥 + 3 − 18𝑥3

−9𝑥4 + 2𝑥 + 1 − (−9𝑥4 + 23𝑥7 + 6𝑥2 − 31)

Multiplying Polynomials

What is the simplified form?


(5x – 3)(2x + 1)

(3x – 4)(x + 2)

(n – 6)(4n – 7)

(2p2 + 3)(2p – 5)

(x + 1)(x + 4)

(x + 2)(x + 4)


𝑥 + 2 𝑥 − 9

(𝑥 + 2)(2𝑥2 − 4𝑥 + 1)

(𝑥3 + 3𝑥 − 4)(2 + 𝑥 − 7𝑥2)

Bell Ringer

1. (𝑥 + 7)(𝑥2 + 8𝑥 − 3)

2. (𝑥 − 3)(𝑥2 − 4𝑥 + 8)

Dividing

Polynomials

Algebra 2

Dividing Polynomials: Box

Method

𝑥4 − 2𝑥3 − 12𝑥2 + 𝑥 + 2

𝑥2 + 3𝑥 + 1


Method

What is 2𝑥4 − 11𝑥2 − 30𝑥 − 27 divided by 𝑥2 + 2𝑥 + 3 ?


Method

What is 4𝑥3 − 2𝑥2 + 10𝑥 − 9 divided by 𝑥2 − 1 ?


Method

What is 𝑥4 − 4𝑥2 + 2𝑥 − 9 divided by 𝑥2 + 𝑥 + 1 ?

Dividing Polynomials: Synthetic

Division

Can use when the divisor (denominator) looks like x – a or x + a


Division Examples

2𝑥2 + 𝑥 − 10

𝑥 − 2


Division Examples

(𝑏3−2𝑏2 − 12𝑏 − 5) ÷ (𝑏 + 1)


Division Examples

7𝑥3 − 6𝑥 + 9 ÷ (𝑥 + 5)

Factor Theorem

When dividing by x – a, if the remainder is 0,

then x – a is a factor of the polynomial.

Factor Theorem: Examples

Is (x + 3) a factor of 𝑥3 + 3𝑥2 − 4x − 12?

Factor Theorem: Examples

Is (𝑥 + 1) a factor of 𝑥4 − 4𝑥3 − 6𝑥2 + 4𝑥 + 5?

Dividing Polynomials: Homework

– Dividing Polynomials Worksheet

Bell Ringer

Divide the following polynomial:

3𝑥5 − 7𝑥4 − 4𝑥3 + 9𝑥2 − 5𝑥 − 1

𝑥 + 1

Factor the following polynomial:

𝑥3 + 𝑥2 − 3𝑥 − 3

Solving

Polynomials:

Day 1

Algebra 2

Solve the Following Quadratic by

Factoring

𝑥2 − 3𝑥 − 28 = 0

Fundamental Theorem of

Algebra

– The degree of a polynomial tells us the number of

solutions (zeroes or roots) that a polynomial has.

– These solutions (zeroes or roots) can be real or

imaginary.

Solving Polynomial Equations

– Find the zeroes of the following polynomials:

1. 𝑓 𝑥 = (𝑥 + 2)(𝑥 − 1)(𝑥 + 3)

2. 𝑓 𝑥 = (2𝑥 + 3)(4𝑥 − 5)(6𝑥 − 1)

Solving Polynomial Equations:

Factoring

𝑥3 − 4𝑥2 − 5𝑥 + 20 = 0


Factoring

𝑥3 − 64 = 0


Factoring

𝑥3 − 7𝑥2 − 𝑥 + 7 = 0


Factoring

𝑥3 − 729 = 0


Factoring

𝑥3 + 6𝑥2 + 25𝑥 = 0


Factoring

𝑥3 + 11𝑥2 − 𝑥 − 11 = 0


Factoring

𝑥3 + 4𝑥2 + 4𝑥 = 0


Factoring Homework

–Solving Polynomial Equations by

Factoring Worksheet

Bell Ringer

– Solve each of the following polynomial equations:

1. 𝑥3 + 729 = 0

2. 𝑥3 + 11𝑥2 − 𝑥 − 11 = 0

Solving

Polynomials:

Day 2

Algebra 2


When We Can’t Factor

– Find the root(s) of the following polynomial equation:

𝑥3 + 2𝑥2 − 19𝑥 − 20 = 0



– Find the solution(s) to the following polynomial equation:

𝑥3 + 3𝑥2 − 13𝑥 − 15 = 0



– Find the zero(s) of the following polynomial equation:

𝑥3 + 4𝑥2 + 6𝑥 + 4 = 0



– Find the root(s) of the following polynomial equation:

𝑥3 + 10𝑥2 + 30𝑥 + 28 = 0


Factoring Homework

–Solving Polynomial Equations

Worksheet

Solving

Polynomial

Equations

Day 3

Find All Roots

2. 𝑥3 − 𝑥2 − 44𝑥 − 70 = 0

Find All Roots

5. 𝑥3 + 5𝑥2 − 13𝑥 + 7 = 0

Find All Roots

7. 𝑥3 + 13𝑥2 + 39𝑥 − 5 = 0

Find All Roots

8. 𝑥3 + 13𝑥2 + 23𝑥 + 11 = 0

Find All Roots

9. 𝑥3 − 11𝑥2 + 37𝑥 − 39 = 0

Find All Roots

10. 𝑥3 + 𝑥2 − 20𝑥 − 50 = 0

Find All Roots

12. 𝑥3 + 6𝑥2 − 11𝑥 − 10 = 0

Find All Roots

13. 𝑥3 − 4𝑥2 + 5𝑥 − 2 = 0

Find All Roots

14. 𝑥3 − 3𝑥2 − 12𝑥 + 10 = 0

– Find all zeroes

𝑥3 + 3𝑥2 − 19𝑥 + 3 = 0

Bell Ringer

Rational

Expressions &

Equations

Algebra 2

– What is a polynomial?– A polynomial is an expression of algebraic terms (has letters and #’s)

– What is a rational expression?– Think of what makes a rational number a rational number…

– A rational expression is a polynomial being divided by another polynomial

– The key to rational expressions is being able to break them down….

Rational Expressions

– What is Domain?

– Is the domain of a rational function restricted

(doesn’t exist somewhere)?

– Think about what you know about rational

expressions…

Rational Expressions: Domain

Rational Expressions:

Simplifying & Excluded Values

–𝑥2+𝑥−90

𝑥2+2𝑥−80–

𝑣2−6𝑣−40

𝑣2−18𝑣+80

Rational Expressions:

Simplifying & Ex. Values

–6𝑘3+30𝑘2

6𝑘3+36𝑘2+30𝑘–

𝑡2−49

𝑡2−14𝑡+49

–20

24=

𝑥

36

How would you solve this?

–𝑦−3

𝑦+4=

5

7

How would you solve this then?

–1+6𝑥

𝑥2+10𝑥=

5

𝑥2+10𝑥

Simple Rational Equations

–12

𝑝−12=

13

𝑝+5

Simple Rational Equations

– Rational Expressions & Equations

Homework

Rational Expressions &

Equations: Homework

solving systems of equations - ms. griggs · 2019-10-27 · solving systems of equations. bell...

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