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### Quotient of polynomial using (Synthetic Division) and Zeros of Polynomial Functions using (Factor Theorem)

• 2. STEPS FOR SYNTHETIC DIVISION METHOD (For divisor x + c)  Write the numerical coefficients of the dividend in the 1st row.  On one side of the first row, write the additive inverse of the constant of the divisor (c). This will serves as the multiplier later.  Bring down the first term on the third row.  Multiply the value in step 3 by c. Write the product on the 2nd row of the next column.  Add the result in step 4 with the number on the same column and write the sum on the 3rd row.  Repeat steps 4 and 5 until the last coefficient, and until the bottom row is complete.  The numbers on the third row serve as the numerical coefficient, of the quotient. To get the literal coefficients, and their degree, just abstract one from the highest degree and affix them with the constant arranged in descending order.
• 3. NOTE: For divisor(ax+ c), the same procedures are to beb followed. But the multiplier will now be 𝑥 = 𝑐 𝑎 . After getting the quotient, we divide the quotient by a, which is the numerical coefficient of x in the divisor.
• 4. Examples: Find the quotient when x³-9x²+23x-15 is divided by x-3. Row 1 1 -9 23 -15 ˪𝟑 Row 2H3 -1815 _______________________ Row 3 1 -6 5 0 Therefore, the quotient is x²-6x-5.
• 5. Example: Find the quotient when 3x⁴-4x³+11x²+7x-2 isdividedby3x+2. Row 1 3-4117 -2 ˪ − 𝟐 𝟑 Row 2H-2 410 2 _____________________________ Row 3 3 -6 15-3 0 The quotient after performing synthetic division i 3x³-6²+15x-3. We will divide this quotient by the numerical coefficient of x in the divisor which is 3. Therefore, the final answer is x³-2x²+5x-1.
• 6. Example:3 Find the quotient when x³-4x²+5x+2 isdividedbyx+1 Row 1 1 -4 5 2˪ − 𝟏 Row 2H-1 5-10 __________________________ Row 3 1-510-8 Affixing all the literal coefficients, we can have x²-5x+10. Notice that after the constant, 10, there is still another constant, -8. It only means that his value serves as the remainder of the two polynomials after dividing. Therefore, we write the quotient of the two polynomials as, x²-5x+10- 𝟖 𝒙+𝟏 .
• 7.  Reminder: Steps in dividing polynomials with divisors x²+bx+c: 1.Write the coefficients of the divisor in Row 1. 2.Take the additive inverse of b and c, and write them on one side of Row 1. 3.Bring down the first coefficient on Row 4. 4.Multiply the number in Row 4 to –b, then write the product on the next column on Row 3, and to –c, then write the product on the next column on Row 2. 5.Add the numbers in column 2 and write the sum in Row 4. 6.Repeat steps 4 and 5 until the last column.
• 9. In finding the zeros of polynomial functions, we are going to apply the previous concepts learned, such as factoring, synthetic division, and remainder theorem more specifically, if the given polynomial is higher than second degree. Always take note that the number of zeros of a polynomial depends on its degree. It means, if the degree of the polynomial is 3, the number of zeroes is also 3, and so on.
• 10. Example: 1. find the zeros of the polynomial function, f(x)= x³-x²-12x f(x)=x³-x²-12x =x³-x²-12x = 0 =x(x²-x-12)= 0 =x(x-4)(x+3)=0 x= 0 x-4 = 0 x+3 = 0 x= 4 x= -3
• 11. Checking: f(x)=x³-x²-12x f(0)= (0)³-(0)²-12(0) f(-3)= (-3)³-(-3)²-12(-3) = 0 = -27- 9 + 36 = -36 + 36 = 0 f(4) = (4)³-(4)²-12(4) = 64- 16- 48 = 64 – 64 = 0 Therefore, we can say that the zeros of the polynomial function, f(x)= x³-x²- 12 are 0, 0 and 0.
• 12. Zeros of Polynomial Functions by Synthetic division
• 13. Example: 1.Find the zeros of the polynomial function, f(x)= x4 +5x³+5x²-5x-6. P= ±1, ±2, ±3, ±6 1 5 5 -5 -6 L 1 q= ±1 1 6 11 6 p 1 6 11 6 0 q = ±1, ±2, ±3, ±6 1 6 11 6 L-1 -1 -5 -6 1 5 6 0
• 14. x² + 5x + 6 = 0 (x + 3) (x + 2) = 0 x+3 = 0 x + 2 = 0 x = -3 x = -2 Therefore, the zeros are 1, -1, -2, and -3
• 15. Quiz: 1.) 3x² + 12x + 4 and x + 4 2.) f(x) = x³ +5x² + 3x – 8 3.) f(x) = x4 -13x² +36
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