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PREC11 Unit1 notes - PRE-CALCULUS 11 Unit 1 Day 1...

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PRE-CALCULUS 11 Unit 1 – Day 1: POLYNOMIALS REVIEW POLYNOMIALS A monomial is a number, a variable, or a product of a number and variables. A monomial that is just a number is also called a constant . The number multiplied to the variables is called the coefficient . A polynomial is a monomial or a sum of monomials . Each monomial is called a term of the polynomial. A binomial is a polynomial with two terms. A trinomial is a polynomial with three terms. The degree of a monomial is the total number of variable factors in the monomial. It can be calculated by adding the exponents of the variable. example: What is the degree of 2 x 4 ? degree = 4 , 2 x 4 is a 4 th degree monomial. exercise: What is the degree of - 4 x 3 yz 2 ? exercise: What is the degree of 2 3 ? The degree of a polynomial is the largest of the degrees from the terms in the polynomial. exercise: What is the degree of 2 x 4 - 4 x 3 yz 2 + 8 ? Polynomials are usually written so that the terms are in descending order of degree . exercise: Write in descending order of degree 5 x + x 8 - 7 x 11 - x 5 + 3 Polynomials of degree 0 is just a constant. Polynomials of degree 1 are linear . Polynomials of degree 2 are quadratic .
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Unit 1 – Day 1: Polynomial Review Page 2 of 2 Operations with Polynomials Operations with polynomials include: Addition – “combine like terms” exercise: Simplify (5 x 2 + x - 7) + (6 x 2 - x - 2) Subtraction – “add its opposite” exercise: Simplify (5 x 2 + x - 7) - (6 x 2 - x - 2) Multiplication – “expand by multiplying every term from the first polynomial to every term in the second polynomial” exercise: Simplify (2 x - 3)( x + 2) Multiplication – “there are some common binomial multiplications that can be simplified quickly” o ( a + b )( a - b ) = a 2 - b 2 exercise: Simplify (3 x + 5)(3 x - 5) o ( a + b ) 2 = a 2 + 2 ab + b 2 exercise: Simplify (3 x + 5) 2 o ( a - b ) 2 = a 2 - 2 ab + b 2 exercise: Simplify (3 x - 5) 2 exercise: Simplify 2(4 x 2 - 3 x + 5) - 3(6 x 2 - 2 x - 1) [answer: - 10 x 2 + 13] exercise: Simplify ( x 2 + 2 x - 3)(4 x 2 - x - 2) [answer: 4 x 4 + 7 x 3 - 16 x 2 - x + 6] exercise: Simplify 5( x - 3) 2 [answer: 5 x 2 - 30 x + 45] x -1 x x 2 x -1 -1 1 1 x x x x -1 -1 -1 -1 -1 -1 - x - x - x x 2
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PRE-CALCULUS 11 Unit 1 - Day 1: Polynomial Review Exercises Simplify. 1. (2 x + y ) + (3 x - 4 y ) 2. (5 x 2 - 3 x + 4) + (3 x 2 - x - 1) 3. (5 a 2 - a - 2) - (4 a 2 + 5 a + 6) 4. (2 m 2 + 2 mn - n 2 ) - ( m 2 - mn - 2 n 2 ) Multiply. 5. (3 xy 2 )( - 4 x 3 y 2 ) 6. ( - 4 rs 3 t 2 )( - 6 rst 4 ) Simplify. 7. 2 3 3 20 5 a b c ab c - 8. 3 4 2 3 36 9 m n p m np - - Expand and simplify. 9. 3( x - 4) + 5( x + 6) 10. 6( a + 3) - 2( a - 5) 11. 2 t (3 t - 4) + t (2 t + 5) 12. 3 y ( y 2 - y - 1) - y (2 y 2 - 3 y + 4) 13. a) Write a 2-digit number in which the units digit is larger than the tens digit. Subtract the tens digit from the units digit and multiply the result by 9. Add the product to the original number. How does the resulting number compare to the original number?
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