Mathematics Extension 1
1. The question consists of various parts related to algebraic expressions and their manipulations.
a) Factorise the expression $x^2 ... show full transcript
Worked Solution & Example Answer:Mathematics Extension 1
1 - HSC - SSCE Mathematics Extension 1 - Question 1 - 2018 - Paper 1
Step 1
a) Factorise the expression $x^2 - 9$ and state the roots of the equation $x^2 - 9 = 0$.
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Answer
To factor the expression x2−9, we recognize it as a difference of squares. Thus:
x2−9=(x−3)(x+3)
To find the roots of the equation x2−9=0, we set the factors equal to zero:
x−3=0⇒x=3
x+3=0⇒x=−3
Therefore, the roots are x=3 and x=−3.
Step 2
b) Solve the equation $2x + 3 = 9$ and provide the solution set.
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Answer
To solve the equation 2x+3=9, we first isolate x:
Subtract 3 from both sides:
2x=9−32x=6
Divide both sides by 2:
x=26x=3
The solution set is therefore {3}.
Step 3
c) Discuss the significance of zero in polynomial equations, particularly in relation to the roots found in part (a).
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Answer
In polynomial equations, the value of zero plays a crucial role as it indicates the points where the polynomial intersects the x-axis, known as the roots. For the factored expression ((x - 3)(x + 3) = 0), the roots x=3 and x=−3 mean that these are the values of x where the expression evaluates to zero.
Thus, the significance of zero is that it identifies values for which the polynomial takes no value (i.e., it is zero), which is essential for understanding the behavior of the polynomial function.
