Y to the power 3 minus 2 Y to the power 2 minus29 Y - 42 factorisee
Answers
Step-by-step explanation:
y3-2y2-29y-42
Final result :
(y + 3) • (y + 2) • (y - 7)
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(((y3) - 2y2) - 29y) - 42
Step 2 :
Checking for a perfect cube :
2.1 y3-2y2-29y-42 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: y3-2y2-29y-42
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -29y-42
Group 2: y3-2y2
Pull out from each group separately :
Group 1: (29y+42) • (-1)
Group 2: (y-2) • (y2)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Polynomial Roots Calculator :
2.3 Find roots (zeroes) of : F(y) = y3-2y2-29y-42
Polynomial Roots Calculator is a set of methods aimed at finding values of y for which F(y)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers y which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 1 and the Trailing Constant is -42.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,6 ,7 ,14 ,21 ,42
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -16.00
-2 1 -2.00 0.00 y+2
-3 1 -3.00 0.00 y+3
-6 1 -6.00 -156.00
-7 1 -7.00 -280.00
-14 1 -14.00 -2772.00
-21 1 -21.00 -9576.00
-42 1 -42.00 -76440.00
1 1 1.00 -72.00
2 1 2.00 -100.00
3 1 3.00 -120.00
6 1 6.00 -72.00
7 1 7.00 0.00 y-7
14 1 14.00 1904.00
21 1 21.00 7728.00
42 1 42.00 69300.00
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
y3-2y2-29y-42
can be divided by 3 different polynomials,including by y-7
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : y3-2y2-29y-42
("Dividend")
By : y-7 ("Divisor")
dividend y3 - 2y2 - 29y - 42
- divisor * y2 y3 - 7y2
remainder 5y2 - 29y - 42
- divisor * 5y1 5y2 - 35y
remainder 6y - 42
- divisor * 6y0 6y - 42
remainder 0
Quotient : y2+5y+6 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring y2+5y+6
The first term is, y2 its coefficient is 1 .
The middle term is, +5y its coefficient is 5 .
The last term, "the constant", is +6
Step-1 : Multiply the coefficient of the first term by the constant 1 • 6 = 6
Step-2 : Find two factors of 6 whose sum equals the coefficient of the middle term, which is 5 .
-6 + -1 = -7
-3 + -2 = -5
-2 + -3 = -5
-1 + -6 = -7
1 + 6 = 7
2 + 3 = 5 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 2 and 3
y2 + 2y + 3y + 6
Step-4 : Add up the first 2 terms, pulling out like factors :
y • (y+2)
Add up the last 2 terms, pulling out common factors :
3 • (y+2)
Step-5 : Add up the four terms of step 4 :
(y+3) • (y+2)
Which is the desired factorization
Final result :
(y + 3) • (y + 2) • (y - 7)
Answer:
first we have take 1
then apko aata hai toh karlena